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