Therapeutic Allogeneic Lymphocytes and Aldesleukin in Treating Patients With High-Risk or Recurrent Myeloid Leukemia After Undergoing Donor Stem Cell Transplant

Inclusion Criteria: * Patients undergoing allogeneic hematopoietic stem cell transplantation for chronic myelogenous leukemia (CML) in accelerated or blast phase, acute myeloid leukemia (AML) beyond first remission, primary refractory AML, therapy-related AML at any stage, or acute leukemia at any stage arising in a patient with an antecedent diagnosis of a myelodysplastic or myeloproliferative syndrome (including chronic myelomonocytic leukemia, CML, polycythemia vera, essential thrombocytosis, and agnogenic myeloid metaplasia with myelofibrosis) * Patients and donors must both be human leukocyte antigen (HLA)-A2 positive * Patients must be able to provide blood and bone marrow samples required for this protocol * Eligibility for Prophylactic Treatment with CD8+ CTL After Transplant (Highest Risk Subgroup): * At time of planned treatment, CD8+ CTL specific for PR3 must have been generated and have completed Quality Control (QC) testing * Patients must have had \> 5% morphologic blasts detectable in bone marrow or peripheral blood just prior to or at the time of transplant * Patients must have evidence of posttransplant recovery of normal hematopoiesis (absolute neutrophil count \[ANC\] \> 500/mm\^3) for at least 7 days prior to the initiation of CTL infusions * Patients on immunosuppressive therapy for graft-versus-host disease (GVHD) are eligible for treatment if not receiving corticosteroids or if the dose of corticosteroids can be tapered to \< the equivalent of 0.5 mg/kg/day of prednisone; The patient's symptoms have to remain stable and unlikely to increase to stage III or IV acute GVHD or chronic GVHD is unlikely to progress following the change in immunosuppressive therapy, after an appropriate monitoring period, as deemed by the patients treating physician and the principal investigator * Eligibility for Treatment with CD8+ CTL at the Time of Relapse After Transplant (All Others): * At time of planned treatment, CD8+ CTL specific for PR3 must have been generated and have completed Quality Control (QC) testing * Patients must have evidence of recurrent/progressive disease posttransplant * Morphologic relapse defined as one or more of the following: abnormal peripheral blasts in absence of growth factor therapy, abnormal bone marrow blasts \> 5% of nucleated cells, extramedullary chloroma or granulocytic sarcoma * Flow cytometric relapse defined as the appearance in the peripheral blood or bone marrow of cells with an abnormal immunophenotype detected by flow cytometry that is consistent with leukemia recurrence/progression * Cytogenetic relapse/progression defined as the appearance in one or more metaphases from bone marrow or peripheral blood cells of either a non-constitutional cytogenetic abnormality identified in at least one cytogenetic study performed prior to transplant or a new abnormality known to be associated with leukemia; (for CML), an increase in the number of Ph+ metaphases from bone marrow or peripheral blood between two consecutive samples after engraftment, or an increase in the percentage of BCR/ABL+ cells by fluorescence in situ hybridization (FISH) between two consecutive samples after engraftment * Molecular relapse/progression defined as a polymerase chain reaction (PCR) assay of bone marrow (BM) or peripheral blood mononuclear cells (PBMC) positive for the presence of the BCR/ABL messenger ribonucleic acid (mRNA) fusion transcript that quantitatively increases by greater than one order of magnitude on a subsequent sample * Patients on immunosuppressive therapy for GVHD at the time of relapse are eligible for treatment if not receiving corticosteroids or if the dose of corticosteroids can be tapered to \< the equivalent of 0.5 mg/kg/day of prednisone; the patient's symptoms have to remain stable and unlikely to increase to stage III or IV acute GVHD or chronic GVHD is unlikely to progress following the change in immunosuppressive therapy, after an appropriate monitoring period, as deemed by the patient's treating physician and the principal investigator Exclusion Criteria: * Exclusions for Treatment at the Time of Relapse/Progression After Transplant: * Patients for whom CD8+ CTL clones specific for PR3 have not been generated by the time of disease relapse/progression post-transplant; these patients can potentially be treated later if CTL become available; patients whose malignant cells do not overexpress PR3, based on direct analysis of a bone marrow sample with \> 50% blasts or of leukemia cells isolated for expression analysis; in either case, patients will be informed about the availability of other treatment protocols for which they might be eligible * Patients with Karnofsky performance status or Lansky play score =\< 30% * Patients with current stage III or IV GVHD unresponsive to therapy or requiring therapy with anti-CD3 mAb, prednisone \> 0.5 mg/kg/day (or corticosteroid equivalent), or other treatments resulting in the ablation or inactivation of T cells (such as other anti-T cell monoclonal antibodies); although the concurrent use of cyclosporine, FK506, or mycophenolate mofetil (MMF) is not strictly an exclusion criterion, attempts should be made to discontinue it if possible * Patients requiring concurrent therapy with hydroxyurea or other agents that may interfere with the function or survival of infused CTL clones * Patients with a preexisting nonhematopoietic organ toxicity that is deemed by the principal investigator to place the patient at unacceptable risk for treatment on the protocol * Patients with graft rejection or failure