Expanded Cord Blood Cell Infusion Following Combination Chemotherapy in Younger Patients With Relapsed or Refractory Acute Myeloid Leukemia

Inclusion Criteria:

* Patients must have a diagnosis of AML or acute leukemia of ambiguous lineage according to World Health Organization (WHO) classification with \>= 5% of disease in bone marrow (BM)
* Recipients of prior allogeneic hematopoietic stem cell transplantation for AML or acute leukemia of ambiguous lineage are eligible if they do not have graft-versus-host disease (GVHD) or they have quiescent GVHD whether or not they are receiving immunosuppressive therapy
* Must have a Lansky or Karnofsky performance status of \>= 50; use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age
* Patients must have recovered from the acute toxicity of all prior chemotherapy
* The following amounts of time must have elapsed prior to entry on study:

  * 2 weeks from local radiation therapy (XRT)
  * 8 weeks from prior craniospinal or if \> 50% of the pelvis has been irradiated
  * 6 weeks must have elapsed if other bone marrow radiation has occurred
* Adequate cardiac, renal, pulmonary, and hepatic function
* Patient must have a life expectancy of at least 2 months
* Females of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of treatment
* Females of childbearing potential and males should agree to use adequate contraception (barrier method of birth control) prior to study entry and for the duration of study participation

Exclusion Criteria:

* Recipients of prior allogeneic hematopoietic stem cell transplant (HSCT) with active acute or chronic GVHD
* Patients with history of Down's syndrome, Fanconi anemia or other known marrow failure condition
* Patients currently receiving other investigational drugs are not eligible
* Current concomitant chemotherapy, radiation therapy, or immunotherapy other than as specified in the protocol with the exception of intrathecal chemotherapy; this includes the tyrosine kinase inhibitor sorafenib which must not be initiated until patient demonstrates count recovery
* Patients with a systemic fungal, bacterial, viral, or other infection not controlled despite appropriate antibiotics or other treatment; uncontrolled systemic infections require infectious disease consultation for verification
* Patients who are platelet refractory prior to initiation of protocol therapy
* Pregnant or lactating patients
* Any significant concurrent disease, illness, or psychiatric disorder that would compromise patient safety or compliance, interfere with consent, study participation, follow up, or interpretation of study results