Ibrutinib or Idelalisib in Treating Patients With Persistent or Relapsed Chronic Lymphocytic Leukemia, Small Lymphocytic Lymphoma, or Non-Hodgkin Lymphoma After Donor Stem Cell Transplant

Inclusion Criteria: * Patients with diagnoses of CLL/SLL or non-Hodgkin lymphoma (NHL) patients, who meet the criteria of either relapse or progression at any time point after allogeneic HCT or those who experience persistent stable disease or persistent disease with regression between days 28 and 100 post-transplant using standard morphologic, flow cytometric, and/or imaging studies and following the disease response evaluation criteria established by the International Workshop on CLL (IWCLL) for CLL and those following Cheson 2007 criteria for NHL * Patients will then be assigned to one of two cohorts: * Cohort 1 will include patients who have relapsed /progressed within the first 180 days post-transplant and who are still within 3 months from date of progression-relapse * Cohort 2 will include patients who have either i) relapsed/progressed beyond day 180 post-HCT, ii) those with persistent stable disease or persistent disease with regression between days 28-100 after allogeneic HCT, or iii) those who progressed or relapsed within 180 days after HCT but were not started on this protocol within 3 months from date of progression or relapse could also be enrolled under cohort 2 * NOTE: the inclusion of patients with persistent stable or persistent regressing disease in this protocol is not meant to advocate treatment; however, if the attending physician is inclined to offer treatment then these patients would be eligible for this study * Patients must be able to give informed consent * Women of childbearing potential and men who are sexually active must affirm they are practicing a highly effective method of birth control during and after the study consistent with local regulations regarding the use of birth control methods for subjects participating in clinical trials; men must agree to not donate sperm during or after the study; for females, these restrictions apply for 1 month after the last dose of study drug; for males, these restrictions apply for 3 months after the last dose of the study drug * Women of childbearing potential must have a negative serum (beta-human chorionic gonadotropin \[B-hCG\]) or urine pregnancy test at screening * Absolute neutrophil count (ANC) \>= 750/mm\^3 * Platelets \>= 30,000/mm\^3 * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x upper limit of normal (ULN) * Total bilirubin =\< 1.5 x ULN unless bilirubin rise is due to Gilbert's syndrome or of non-hepatic origin * Creatinine clearance (Clcr) \> 25 mL/min Exclusion Criteria: * Pregnant or breast feeding females; (lactating females must agree not to breast feed while taking ibrutinib or idelalisib) * Any condition, including the presence of laboratory abnormalities, which places the subject at unacceptable risk if he/she were to participate in the study or confounds the ability to interpret data from the study should be first discussed and clarified with the study investigators * Concurrent use of other anti-cancer agents or treatments * Known history of human immunodeficiency virus (HIV) * Karnofsky performance status \< 50% * Active grades III or IV acute GVHD * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Vaccinated with live, attenuated vaccines within 4 weeks of initiation of therapy * Patients with other prior malignancies except for adequately treated basal cell carcinoma, squamous cell carcinoma of the skin, breast or cervical cancer in situ, or other cancer from which the patient has been disease-free for 5 years or greater, unless approved by the protocol principal investigators * Unable to swallow capsules or disease significantly affecting gastrointestinal function and/or inhibiting small intestine absorption such as; malabsorption syndrome, resection of the small bowel, or poorly controlled inflammatory bowel disease affecting the small intestine * Uncontrolled active systemic fungal, bacterial, viral, or other infection (defined as exhibiting ongoing signs/symptoms related to the infection and without improvement, despite appropriate antibiotics or other treatment) * Have uncontrolled hepatitis B or C infection * IBRUTINIB-SPECIFIC EXCLUSION CRITERIA * History of stroke or intracranial hemorrhage within 6 months of screening would be exclusion for ibrutinib therapy but idelalisib would be an option * Patients requiring anticoagulation with warfarin or equivalent vitamin K antagonists (e.g., phenprocoumon) within 28 days from the start of study drug cannot be treated with ibrutinib but idelalisib would be an option * Patients requiring chronic treatment with strong cytochrome P450 family 3, subfamily A (CYP3A) inhibitors cannot be treated with ibrutinib but idelalisib would be an option * Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 (moderate) or class 4 (severe) cardiac disease as defined by the New York Heart Association Functional Classification would be exclusion for ibrutinib therapy but idelalisib would be an option * IDELALISIB-SPECIFIC EXCLUSION CRITERIA * Ongoing drug-induced liver injury, chronic active hepatitis C (HCV), chronic active hepatitis B (HBV), alcoholic liver disease, non-alcoholic steatohepatitis, primary biliary cirrhosis, extrahepatic obstruction caused by cholelithiasis, cirrhosis of the liver, or portal hypertension would be exclusion for idelalisib therapy but ibrutinib would be an option * Ongoing drug-induced pneumonitis would be exclusion for idelalisib therapy but ibrutinib would be an option