Subcutaneous Recombinant Human IL-15 (s.c. rhIL-15) and Alemtuzumab for People With Refractory or Relapsed Chronic and Acute Adult T-cell Leukemia (ATL)

* INCLUSION CRITERIA: Inclusion Criteria * Age greater than or equal to 18 years; no upper age limit. * Patients diagnosed with a leukemia or lymphoma as follows: * Chronic or acute leukemia forms of Human T-cell lymphotropic virus type 1 (HTLV-1) associated adult T-cell leukemia; * Peripheral T-cell lymphoma (angioimmunoblastic, hepatosplenic, or not otherwise specified); or, * Cutaneous T-cell lymphoma stage III or IV with circulating monoclonal cells (B1 or B2) and/or erythrodermia (T4) * T-cell prolymphocytic leukemia (T-PLL) NOTE: Diagnosis must be validated by the Pathology Department, National Cancer Institute (NCI). -Patients must have measurable or evaluable disease. NOTE: All patients with greater than 10% abnormal cluster of differentiation 4 (CD4+) homogeneous cluster of differentiation 3 (CD3) low strongly cluster of differentiation 25 (CD25+) expressing cells, or greater than 5% Szary cells/T-PLL, among the peripheral blood mononuclear cells (PBMCs) in the peripheral blood will be deemed to have evaluable disease. * Abnormal T cells must be cluster of differentiation 52 (CD52+) as assessed by flow cytometry or immunohistochemistry. * Patients must have a life expectancy of greater than or equal to 2 months. * Patients must have been refractory or relapsed following front line therapy for Adult T-cell Leukemia (ATL); those with cutaneous T-cell lymphoma (CTCL) or peripheral T-cell lymphoma (PTCL) who have cluster of differentiation 30 (CD30+) disease must have progressed during or after treatment with brentuximab vedotin, or are unable to receive treatment due to allergy or intolerance. * Patients must have recovered to less than grade 1 or to baseline from toxicity of prior chemotherapy or biologic therapy and must not have had major surgery, chemotherapy, radiation or biologic therapy within 2 weeks prior to beginning treatment. NOTE: Exceptions to this include events not considered to place the subject at unacceptable risk of participation in the opinion of the PI (e.g., alopecia). * Carbon monoxide diffusing capacity alveolar volume (DLCO/VA) and forced expiratory volume (FEV) 1.0 \> 50% of predicted on pulmonary function tests. * Adequate laboratory parameters, as follows: * Serum creatinine of less than or equal to 1.5 x the upper limit of normal * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 3 x the upper limit of normal * Absolute neutrophil count greater than or equal to 1,500/mm\^3 and platelets greater than or equal to 100,000/mm\^3. * Eastern Cooperative Oncology Group (ECOG) less than or equal to 1. * Patients must be able to understand and sign an Informed Consent Form. * All patients must use adequate contraception during participation in this trial and for 4 months following completing therapy. EXCLUSION CRITERIA: * Patients who have received any systemic corticosteroid therapy within 4 weeks prior to the start of therapy, or 12 weeks if given to treat graft versus host disease (GVHD), with the exception of physiological replacement doses of cortisone acetate or equivalent. * Patients who have undergone allogeneic stem cell transplantation and have required systemic treatment for GVHD (including but not limited to oral or parenteral corticosteroids, ibrutinib, and extracorporeal phototherapy) within the last 12 weeks * Clinical evidence of (parenchymal or meningeal) central nervous system (CNS) involvement or metastasis. In subjects suspected of having CNS disease, a magnetic resonance imaging (MRI) scan of the brain and lumbar puncture should be done to confirm. * Documented human immunodeficiency virus (HIV), active bacterial infections, active or chronic hepatitis B, hepatitis C. * Positive hepatitis B serology indicative of previous immunization (i.e., hepatitis B surface antigen (HBsAb) positive and hepatitis B core antibody (HBcAb) negative) or a fully resolved acute hepatitis B infection is not an exclusion criterion. * If hepatitis C antibody test is positive, then the patient must be tested for the presence of hepatitis C virus (HCV) by reverse transcription polymerase chain reaction (RT-PCR) and be HCV ribonucleic acid (RNA) negative NOTE: HIV-positive patients are excluded from the study. Alemtuzumab may produce a different pattern of toxicities in patients with HIV infection; in addition, the depletion of T cells produced by alemtuzumab may have adverse effects on HIV-positive individuals. * Concurrent anticancer therapy (including other investigational agents). * History of severe asthma or presently on chronic inhaled corticosteroid medications (patients with a history of mild asthma not requiring corticosteroid therapy are eligible). * Patients with smoldering and lymphomatous ATL. * Pregnant or nursing patients. * Patients who have previously received alemtuzumab are ineligible. NOTE: Patients with relapsed T-cell prolymphocytic leukemia (T-PLL) who have achieved at least a partial response to prior alemtuzumab are eligible. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, moderate/severe graft versus host disease, cognitive impairment, active substance abuse, or psychiatric illness/social situations that, in the view of the Investigator, would preclude safe treatment or the ability to give informed consent and limit compliance with study requirements.