Umbilical Cord Blood-Derived Natural Killer Cells, Elotuzumab, Lenalidomide, and High Dose Melphalan, Followed by Stem Cell Transplant in Treating Patients With Multiple Myeloma

Inclusion Criteria:

* Patients with high risk multiple myeloma who are transplant candidates, in partial response (PR) or better; high risk will be defined as patients with any of the following:

  * Fluorescence in situ hybridization showing t(4:14), t(14:16), t(14:20), gain (amp) 1q; deletion (Del) 17/17p \[or tp53 gene mutation/deletion by next generation sequencing (NGS), or by conventional cytogenetics\];
  * Deletion 13 by conventional cytogenetic analysis;
  * High risk signatures as determined by the GEP-70 or EMC-92 gene expression profiles;
  * Relapsed disease within 18 months of prior autologous stem cell transplant (ASCT)
* Patients with plasma cell leukemia who are transplant candidates
* Performance score of at least 70% by Karnofsky or 0 to 2 Eastern Cooperative Oncology Group (ECOG)
* Left ventricular ejection fraction greater than 40%
* Pulmonary function test (PFT) demonstrating a diffusion capacity of least 40% predicted
* Estimated serum creatinine clearance \>= 60 ml/min (using the Cockcroft-Gault formula and/or serum creatinine =\< 1.6 mg/dL
* Serum glutamate pyruvate transaminase (SGPT) less than 3 x upper limit of normal
* Total bilirubin less than 2 x upper limit of normal
* All study participants must be registered into the mandatory Revlimid Risk Evaluation and Mitigation Strategy (REMS) program, and be willing and able to comply with the requirements of the Revlimid REMS program
* Females of reproductive potential must adhere to the scheduled pregnancy testing as required in the Revlimid REMS program
* Men must agree to use a latex condom during sexual contact with females of child bearing potential even if they have had a successful vasectomy
* Patients must have a cord blood (CB) unit available which is matched with the patient at 4, 5, or 6/6 human leukocyte antigen (HLA) class I (serological) and II (molecular) antigens
* Patient or legally authorized representative able to sign informed consent

Exclusion Criteria:

* Patients receiving any other investigational agents
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to melphalan
* Known hypersensitivity or desquamating rash to either thalidomide or lenalidomide
* Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, uncontrolled hypertension (systolic \> 160, diastolic \> 100 despite antihypertensive therapy, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Human immunodeficiency virus (HIV)-positive patients are excluded due to increased risk of lethal infections when treated with myeloablative chemotherapy