Chemotherapy, Total-Body Irradiation, Donor Natural Killer Cell Infusion, Aldesleukin, and UCB Transplant in Treating Patients With Relapsed or Refractory AML

Inclusion Criteria: * Aged 0 to 45 years who meet one of the following criteria: * Primary induction failure defined as no complete remission (CR) after two or three induction cycles (no blast limit). * Relapsed acute myeloid leukemia (AML) with low disease burden * For patients 19 through 45 years of age: must have less than 10% marrow blasts at time of enrollment for patients who did not receive re-induction or measured at least 28 days from the start of re-induction therapy. Patients who have relapsed more than 12 months following a prior hematopoietic cell transplant (HCT) and did not reach CR following one re-induction cycle but have less than 10% marrow blasts are eligible. * For patients 0 through 18 years of age: must have less than 50% marrow blasts after no more than 3 induction attempts * CR3 or greater. This will include CRp defined as CR without platelet recovery to 100,000/mcL. * CR1 or CR2 with high risk features (therapy induced, prior myelodysplastic syndrome (MDS) or myeloproliferative disease (MPD), high risk cytogenetic or molecular phenotype) with no available alternate (sibling, URD or UCB) donors. * Patients with prior central nervous system (CNS) involvement are eligible provided that it has been treated and is in remission. CNS therapy (chemotherapy or radiation) should continue as medically indicated during the protocol. * Have acceptable organ function within 14 days of enrollment defined as: * Renal: creatinine ≤ 2.0 mg/dL (adult patients) or calculated creatinine clearance \> 40 ml/min (pediatric patients) * Hepatic: bilirubin, AST/ALT, ALP ≤ 5 x upper limit of normal * Pulmonary function: DLCOcorr \> 50% of normal, (oxygen saturation \[\>92%\] can be used in child where PFT's cannot be obtained) * Cardiac: left ventricular ejection fraction ≥ 45% * Karnofsky score (adults) \> 70% or Lansky score \> 50% (pediatrics) * Women of childbearing potential must agree to use adequate contraception (diaphragm, birth control pills, injections, intrauterine device \[IUD\], surgical sterilization, subcutaneous implants, or abstinence, etc.) for the duration of treatment. * All patients will be questioned about prior exposure to antibody therapy (including OKT3, rituximab, trastuzumab, and gemtuzumab) without affect to eligibility. Patients with prior exposure will have a blood sample collected for human anti-mouse antibody (HAMA). For patients with no prior antibody therapy exposure, no further action will be taken. * Not receiving prednisone or other immunosuppressive medications * Voluntary written consent Exclusion Criteria: * Active infection at time of enrollment or documented fungal infection within 3 months * Evidence of HIV infection or known HIV positive serology * Pregnant or breast feeding. The agents used in this study may be teratogenic to a fetus and there is no information on the excretion of agents into breast milk. All females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy. * If ≤ 18 years old, prior myeloablative transplant within the last 6 months. If \> 18 years old prior myeloablative allotransplant or autologous transplant * Extensive prior therapy including \> 12 months of any alkylator chemotherapy (etoposide \>100 mg/m\^2 x 5 days, cyclophosphamide \>1 gm/m\^2 or mitoxantrone \>8 gm/m\^2) delivered at 3-4 week intervals or \> 6 months alkylator therapy (as above) with extensive radiation (determined by Radiation Oncology, e.g. mantle irradiation for Hodgkin's) and/or prior radiation therapy that makes a patient ineligible for total body irradiation (TBI). Criteria for Second Course of IL-2 (begin day +60): * No Graft-Versus-Host Disease (GVHD), active infection or any other severe medical co-morbidity * Absolute neutrophil count (ANC) \> 1000 without growth factor support * No grade 4 toxicity (except fevers) attributed to IL-2 during course #1