Inclusion Criteria:

* Age 18 or older and able to swallow pills
* Diagnosis of T-LGLL defined as: CD3+CD8+ cell population \> 650/mm\^3 or CD3+CD8+CD57+ population \> 500/mm\^3 and the presence of a clonal T-cell receptor (within 1 month of diagnosis). Note: patients with MDS-like T-LGLL may be included with PI approval even if CD3+CD8+ cell population is \< 650/mm\^3, though +TCR is required. Natural-Killer (NK) LGL is also permitted, provided there is a clonal NK-cell population noted with \> 500 cells/mm\^3
* Untreated T-LGLL or failed at least one line of frontline therapy;
* Patients must be off treatment for at least 14 days or 5 half-lives, whichever is longer
* Require Treatment for T-LGLL (one or more required)

* Symptomatic anemia with hemoglobin \< 10 g/dL
* Transfusion-dependent anemia
* Neutropenia with absolute neutrophil count (ANC) \< 500/mm\^3
* Neutropenia with ANC \< 1500/mm\^3 with recurrent infections
* Platelet count \> 50 x 10\^9/L. Platelet transfusion may be utilized to meet inclusion criteria, as long as the platelet count remains \>50,000/uL within 5 days of last transfusion. Note: Patients with platelets \<100 x 109/L and renal impairment are not permitted to enroll to the study. Renal impairment is defined as creatinine clearance (CrCl) \< 90 mL/min.
* Serum creatinine =\< 2 x the upper limit of normal (ULN)
* - Estimated glomerular filtration rate (eGFR) =\> 30 mL/min using the Modification of Diet in Renal Disease (MDRD) equation (multiplying eGFR by each subjects Body Surface Area \[BSA\])
* Total bilirubin =\< 1.5 x ULN (patients with Gilbert's syndrome with a bilirubin \> 1.5 x ULN permitted)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN
* Alkaline phosphatase (ALP) =\< 2.5 x ULN
* Eastern cooperative oncology group (ECOG) performance status =\< 2
* Men and women of reproductive potential must agree to follow accepted birth control methods for the duration of the study. Female subject is either post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study treatment until 5 half-lives have passed. Male subject agrees to use an acceptable method for contraception for the duration of the study treatment until 5 half-lives have passed.
* Able to sign informed consent

Exclusion Criteria:

* Absolute neutrophil count (ANC) less than 100/mm\^3
* Active infection requiring ongoing anti-microbial treatment. Patients with human immunodeficiency virus (HIV), positive hepatitis B surface antigen or hepatitis C antibody will be excluded
* Concurrent immune-suppressive therapy (prednisone or equivalent up to 20 mg permitted to treat T-LGL symptoms, but must be weaned within one month of initiation of trial drug). Patients on stable, chronic prednisone =\< 10 mg for rheumatologic/autoimmune conditions are exempted from this requirement. They may enroll on the study
* Active, concurrent malignancy unless deemed related to T-LGLL by principal investigator (PI). Early stage skin cancers, prostate cancer, permitted if under no active therapy
* For females of childbearing potential: Positive pregnancy test or lactating
* Unstable angina or myocardial infarction within the past 2 months
* Chronic obstructive pulmonary disease or other interstitial lung disease in active exacerbation
* Cirrhosis