Phase II Study of CD5 CAR Engineered IL15-transduced Cord Blood-derived NK Cells in Conjunction With Lymphodepleting Chemotherapy for the Management of Aggressive T Cell Hematological Malignancies

Inclusion Criteria: 1. 18-80 years of age 2. Patients with hematological malignancies with an expression of CD5 in the tumor sample of ≥ 30% measured by immunohistochemistry or flow cytometry. 3. Patients must meet disease-specific eligibility criteria. a. Patients with a history of T-lymphoid malignancies, defined as T cell acute lymphoblastic leukemia/lymphoma (T-ALL/T-LBL), T-PLL, Peripheral T-cell lymphoma (PTCL-NOS), Hepatosplenic gamma/delta NHL, AITL, Alk negative/ DUSP22 negative ALCL, or other subtypes of T cell NHL with indication for autologous or allogeneic transplant in CR-1 Patients with T-ALL and T-PLL who are in morphologic remission but flow MRD positive are eligible. 4. Patients should be at least 1 week from last cytotoxic chemotherapy at the time of starting lymphodepleting chemotherapy. Patients may continue tyrosine kinase inhibitors or other targeted therapies until at least three days prior to administration of lymphodepleting chemotherapy. 5. Localized radiotherapy to one or more disease sites is allowed. 6. Karnofsky Performance Scale \> 50%. 7. Adequate organ function: 1. Renal: Serum creatinine ≤ 2.0 ULN or estimated Glomerular Filtration Rate (eGFR using the CKI-EPI equation) ≥ 30 ml/min/1.73 m2. 2. Hepatic: ALT/AST ≤ 3.0 x ULN or ≤ 5 x ULN if documented liver involvement with disease, Total bilirubin ≤ 2.0 ULN, except in subjects with Gilbert's Syndrome in whom total bilirubin must be ≤ 3.0 mg/dL. No history of liver cirrhosis. 3. Cardiac: Cardiac ejection fraction ≥ 40%, no clinically significant pericardial effusion as determined by an ECHO or MUGA, and no uncontrolled arrhythmias or symptomatic cardiac disease. 4. Pulmonary: No clinically significant lung involvement, per PI discretion, pleural effusion, baseline oxygen saturation \> 92% on room air. 8. Ability to understand and the willingness to sign a written informed consent document. 9. Weight ≥40 kg. 10. English and non-English-speaking patients are eligible. 11. Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. (Refer to Pregnancy Assessment Policy MD Anderson Institutional Policy # CLN1114). This includes all female patients, between the onset of menses (as early as 8 years of age) and 55 years unless the patient presents with an applicable exclusionary factor which may be one of the following: * Postmenopausal (no menses in greater than or equal to 12 consecutive months). * History of hysterectomy or bilateral salpingo-oophorectomy. * Ovarian failure (Follicle Stimulating Hormone and Estradiol in menopausal range, who have received Whole Pelvic Radiation Therapy). * History of bilateral tubal ligation or another surgical sterilization procedure. Approved methods of birth control are as follows: Hormonal contraception (i.e. birth control pills, injection, implant, transdermal patch, vaginal ring), Intrauterine device (IUD), Tubal Ligation or hysterectomy, Subject/Partner post vasectomy, Implantable or injectable contraceptives, and condoms plus spermicide. Not engaging in sexual activity for the total duration of the trial and the drug washout period is an acceptable practice; however periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of birth control. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 3 months after completion of study therapy. 12. Signed consent to long-term follow-up protocol PA17-0483. Exclusion Criteria: 1. Positive beta HCG in female of child-bearing potential defined as not postmenopausal for 24 months or no previous surgical sterilization or lactating females. 2. Presence of clinically significant Grade 3 or greater toxicity from the previous treatment, as determined by PI. 3. Presence of uncontrolled fungal, bacterial, viral, or other infection not responding to appropriate therapy. 4. Active hepatitis B or C. 5. HIV with detectable viral load. 6. Presence of active neurological disorder(s). 7. Active autoimmune disease within 12 months of enrollment 8. Active cerebral or meningeal involvement by the malignancy 9. Active (defined as requiring therapy) acute or chronic GVHD. 10. Any other malignancy known to be active, except for treated cervical intra-epithelial neoplasia and non-melanoma skin cancer. 11. Presence of any other serious medical condition that may endanger the patient at the investigator criteria. 12. Major surgery \<4 weeks prior to first dose of study drug 13. Allogeneic SCT or DLI \<12 weeks prior to first dose of study drug. Recipients of an allogeneic SCT patients should have discontinued all forms of immunosuppression at least 8 weeks prior to enrollment in the study. 14. Concomitant use of other investigational agents. 15. Concomitant use of other anti-cancer agents. 16. Patients receiving systemic steroid therapy at the time of NK cell infusion (physiological substitutive doses are allowed) or have received ATG or lymphocyte immune globulin within 14 days or alemtuzumab within 3 months of enrollment. 17. Patients receiving immunosuppressive therapy. 18. Patients with diminished mental capacity will not be enrolled in the study.