Targeted Therapy in Treating Patients With Relapsed or Refractory Acute Lymphoblastic Leukemia or Acute Myelogenous Leukemia

Inclusion Criteria:

* Participants \>= 18 years of age with relapsed/refractory leukemia with a confirmed diagnosis of acute myelogenous leukemia (AML) or acute lymphoblastic leukemia (ALL) who meet the following criteria:

  * Individuals aged 18-64 years with salvage treatment failures only - defined as relapsed or refractory to after least 1 cycle of salvage therapy

    * Given the clinical activity and use of hypomethylating agents in AML patients, initial and salvage therapy may include hypomethylating agents
  * Age \>= 65 years: Refractory to induction chemotherapy - defined as no response to initial therapy or have relapsed after initial therapy

    * Individuals aged \>= 65 years, with chronic myelomonocytic leukemia (CMML) or myelodysplasia (MDS) that transform to acute leukemia while actively receiving hypomethylating agents (i.e., decitabine or azacytidine) will be considered induction failures and are thus eligible for this trial; for Philadelphia positive (Ph+) ALL, initial therapy and salvage therapy may include steroids and imatinib or dasatinib
* Primary patient samples must show in vitro kinase inhibitor sensitivity as determined by the Oregon Health and Science University (OHSU) functional kinase inhibitor screen; for OHSU patients, functional kinase inhibitor screening may be performed as part of this study or through enrollment in eIRB4422 if the identical Food and Drug Administration (FDA)/Clinical Laboratory Improvement Act (CLIA) approved assay is used and a result is available within 2 weeks of starting on study drug treatment
* Patients must have normal organ function as defined below:

  * Serum creatinine \< 2.0 x institutional upper limit of normal (ULN)
  * International normalized ratio (INR) \< 1.5 x institutional ULN
* Adequate hepatic function as defined by the following criteria:

  * Total serum bilirubin =\< 1.5 x ULN, unless due to Gilbert's syndrome
  * Alanine aminotransferase (ALT) =\< 2.5 x ULN
  * Aspartate aminotransferase (AST) =\< 2.5 x ULN
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Discontinuation of anti-coagulants and anti-platelet drugs at least 7 days prior to start of study drug

  * Aspirin 81 mg is permitted as long as platelet count is \> 50 and there is no evidence of active bleeding or coagulopathy (INR \> 1.5, fibrinogen \> 150)
* No uncontrolled infections as determined by the investigator
* No clinically significant thyroid disease (e.g. hyperthyroid/hypothyroidism)
* No active graft versus host disease (GVHD): patients with a history of stem cell transplant are eligible but cannot have evidence of active GVHD as determined by the investigator
* Must be able to take oral medication
* Women of childbearing potential must have a negative serum or urine pregnancy test (sensitivity \< 25 IU human chorionic gonadotropin \[HCG\]/L) within 72 hours prior to the start of study drug
* Persons of reproductive potential must agree to use an adequate method of contraception throughout treatment and for at least 4 weeks after study drug is stopped; women of childbearing potential and men with a sexual partner of childbearing potential must be advised of the importance of avoiding pregnancy during trial participation and the potential risk factors for an unintentional pregnancy
* Ability to understand and the willingness to sign a written informed consent and Health Insurance Portability and Accountability Act (HIPPA) document
* Serum sodium (Na), potassium (K), magnesium (Mg), and total serum calcium (Ca) or ionized Ca levels must be greater than or equal to the institutional lower limit of normal; subjects with low K or Mg levels, total corrected serum Ca and/or ionized Ca must be replete for protocol entry
* Dasatinib

  * Discontinuation of any medications known to contribute significantly to the risk of QT prolongation at least 48 hours prior to start of study drug; Levaquin and Zofran are an exception; of note, certain agents that prolong the corrected QT (QTc) may be allowed but only after discussion with the chemotherapy pharmacist; should the investigator believe that therapy with a potentially QT prolonging medication is vital to an individual subject's care, then additional electrocardiogram (ECG)s should be done at the investigator's discretion to ensure the subject's safety
  * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) within 24 hours prior to the start of study drug
  * Men who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug plus 90 days (duration of sperm turnover) for a total of 90 days post-treatment completion
  * Azoospermic males and WOCBP, who are not heterosexually active, are exempt from contraceptive requirements; however, WOCBP must still under pregnancy testing as described in this section
  * Investigators shall counsel WOCBP and male subjects who are sexually active with WOCBP on the importance of pregnancy prevention and the implications of unexpected pregnancy; investigators shall advise WOCBP and male subjects who are sexually active with WOCBP on the use of highly effective contraception; highly effective methods of contraception have a failure rate of \< 1% when used consistently and correctly
  * At a minimum, subjects must agree to the use of two methods of contraception, with one method being highly effective and the other method being either highly effective or less effective
* Sorafenib

  * Creatinine \< 1.5 X ULN
* Ponatinib

  * Female and male patients who are fertile must agree to use an effective form of contraception with their sexual partners from randomization through 4 months after the end of treatment
  * Discontinuation of any medications known to contribute significantly to the risk of QT prolongation at least 48 hours prior to start of study drug; Levaquin and Zofran are an exception; of note, certain agents that prolong the QTc may be allowed but only after discussion with the chemotherapy pharmacist; should the investigator believe that therapy with a potentially QT prolonging medication is vital to an individual subject's care, then additional ECGs should be done at the investigator's discretion to ensure the subject's safety
  * Serum lipase =\< 1.5 x ULN
  * Serum amylase =\< 1.5 x ULN
* Pacritinib

  * Discontinuation of any medications known to contribute significantly to the risk of QT prolongation at least 48 hours prior to start of study drug; Levaquin and Zofran are an exception; of note, certain agents that prolong the QTc may be allowed but only after discussion with the chemotherapy pharmacist; should the investigator believe that therapy with a potentially QT prolonging medication is vital to an individual subject's care, then additional ECGs should be done at the investigator's discretion to ensure the subject's safety

Exclusion Criteria:

* Any leukemia treatment within 1 week (for cytotoxic therapy) and/or 5 half lives (for targeted agents) prior to starting study drug; corticosteroids are allowable throughout the study to treat concomitant medical disorders per provider discretion; hydroxyurea is allowed prior to enrollment and after the start of the study drug for the control of peripheral leukemic blasts in subjects with leukocytosis per physician discretion
* Recent uncontrolled angina, recent \> New York Heart Association (NYHA) class II congestive heart failure, or recent myocardial infarction (MI) within 6 months prior to start of study treatment
* Diagnosed congenital long QT syndrome
* Any recent history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or torsades de pointes)
* History of clinically significant bleeding disorder unrelated to cancer
* Drugs that affect the cytochrome P450 family 3 subfamily A polypeptide 4 (CYP3A4) system (inducers/inhibitors/substrates) are allowed but should be used with caution depending on specific kinase inhibitor used; dietary supplements will be discouraged; however, their use may be allowed on a case by case basis per the discretion of the investigator after consultation with an oncology pharmacist
* Uncontrolled intercurrent illness that would limit compliance with study requirements
* Pregnant or lactating women are excluded from this study
* Known human immunodeficiency virus (HIV)-positive patients are excluded from the study
* History of hypersensitivity to any of the kinase inhibitors included in this study
* Dasatinib

  * Known pulmonary arterial hypertension
  * Patients may not have a clinically significant pleural or pericardial effusion
  * Uncontrolled hypertension: inability to maintain blood pressure below the limit of 140/90 mmHg
  * Any history of second or third degree heart block (may be eligible if the subject currently has a pacemaker)
  * Prolonged QTc interval (\> 450 msec for men and \> 470 msec for women) on pre-entry electrocardiogram
* Sorafenib

  * Major surgery, open biopsy, or significant traumatic injury within 30 days
  * Non-healing wound, ulcer, or bone fracture
  * Thrombotic or embolic venous or arterial events, such as cerebrovascular accident, including transient ischemic attacks, arterial thrombosis, deep vein thrombosis and pulmonary embolism within the past 6 months

    * Line associated deep vein thrombosis (DVTs) which are adequately treated (line removed and/or patient anticoagulated) are permitted
  * Uncontrolled hypertension
  * Active bleeding during screening
* Ponatinib

  * History of acute pancreatitis within 1 year of study or history of chronic pancreatitis
  * QTC \> 450 msec for men and \> 470 msec for women
  * Uncontrolled hypertriglyceridemia (triglycerides \> 450 mg/dL)
  * Any history of myocardial infarction, stroke, or revascularization
  * Any history of venous thromboembolism including deep venous thrombosis or pulmonary embolism
  * Unstable angina or transient ischemic attack within 6 months prior to start of study treatment
  * Congestive heart failure within 6 months prior to enrollment, or left ventricular ejection fraction (LVEF) less than lower limit of normal per local institutional standards within 6 months prior to enrollment
  * History of clinically significant (as determined by the treating physician) atrial arrhythmia
  * Uncontrolled hypertension (diastolic blood pressure \> 90 mm Hg; systolic \> 140 mm Hg); patients with hypertension should be under treatment on study entry to effect blood pressure control
  * History of ongoing alcohol abuse
  * Ocular toxicity present as measure during a comprehensive eye exam
* Pacritinib

  * Major surgery, open biopsy, or significant traumatic injury within 30 days
  * Active bleeding during screening
  * QTC \> 450 mSec for men and \> 470 msec for women
  * NYHA class II congestive heart failure (a history of CHF is allowed as long as this has resolved to \< NYHA class II within 30 days of initiation of pacritinib
* Ruxolitinib

  * Evidence of hepatitis B virus (HBV) or hepatitis C virus (HCV) infection or risk of reactivation: HBV deoxyribonucleic acid (DNA) and HCV ribonucleic acid (RNA) must be undetectable; subjects cannot be positive for hepatitis B surface antigen (HBsAg) or anti-hepatitis B core antibody; subjects who have positive anti-hepatitis B surface antibdy (HBs) as the only evidence of prior exposure may participate in the study provided that there is both 1) no known history of HBV infection, and 2) verified receipt of hepatitis B vaccine
* Idelalisib

  * Ongoing drug-induced liver injury, chronic active hepatitis C (HCV), chronic active hepatitis B (HBV), alcoholic liver disease, non-alcoholic steatohepatitis, primary biliary cirrhosis, extrahepatic obstruction caused by cholelithiasis, cirrhosis of the liver, portal hypertension, or history of autoimmune hepatitis
  * Ongoing symptomatic pneumonitis.
  * Ongoing inflammatory bowel disease or autoimmune colitis.
  * Ongoing cytomegalovirus (CMV) infection, treatment, or prophylaxis within the past 28 days prior to the screening test for active CMV
  * History of serious allergic reaction including anaphylaxis and epidermal necrolysis