Fludarabine Phosphate and Total-Body Irradiation Before Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Chronic Lymphocytic Leukemia or Small Lymphocytic Leukemia

Inclusion Criteria: * Patients with CLL (or small lymphocytic lymphoma) or diagnosis of CLL that progresses to prolymphocytic leukemia (PLL), or T-cell CLL or PLL * Patients with B-Cell CLL or PLL who have at least one of the following: * Failed to meet National Cancer Institute (NCI) Working Group criteria for complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog, e.g. cladribine \[2-CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine (or another nucleoside analog) * Failed FCR combination chemotherapy at any time point * Had de novo of acquired "17p deletion" cytogenetic abnormality; patients should have received induction chemotherapy but could be transplanted in first (1st) complete response (CR) * Patient has a suitable human leukocyte antigen (HLA)-matched related donor who is willing to undergo leukapheresis initially for collection of PBSC and subsequently for collection of peripheral blood mononuclear cells (PBMC) with filgrastim (G-CSF) mobilization and willing to donate stem cells * DONOR: Related donor who is HLA phenotypically or genotypically identical at the allele level at HLA-A, -B, -C, -DRB1, and -DQB1 * DONOR: Donor must consent to G-CSF administration and leukapheresis * DONOR: Donor must have adequate veins for leukapheresis or agree to placement of central venous catheter (femoral, subclavian) Exclusion Criteria: * Infection with human immunodeficiency virus (HIV), human T-lymphotropic virus (HTLV)-1, or HTLV-2 * Active central nervous system (CNS) involvement with CLL * Patients with active non-hematologic malignancies (except non-melanoma skin cancers) * Patients with a history of non-hematologic malignancies (except non-melanoma skin cancers) currently in a complete remission, who are less than 5 years from the time of complete remission, and have a \> 20% risk of disease recurrence * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Pregnant or breastfeeding women * Karnofsky score =\< 70 * Fungal infections with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Cytotoxic agents for "cytoreduction" (with the exception of imatinib mesylate \[Gleevec\], cytokine therapy, hydroxyurea, chlorambucil or Rituxan) within three weeks of the initiation of conditioning * Active bacterial or fungal infections unresponsive to medical therapy * Cardiovascular: cardiac ejection fraction \< 40%; patients with poorly controlled hypertension despite multiple antihypertensives * Pulmonary: diffusing capacity of carbon monoxide (DLCO) \< 40%, total lung capacity (TLC) \< 40%, forced expiratory volume in one second (FEV1) \< 40% and/or requiring continuous supplementary oxygen, or severe deficits in pulmonary function testing as defined by pulmonary consultant service * Liver function abnormalities: patients with clinical or laboratory evidence of liver disease would be evaluated for the cause of liver disease, its clinical severity in terms of liver function, bridging fibrosis, and the degree of portal hypertension; patients will be excluded if they are found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3mg/dL, or symptomatic biliary disease * DONOR: Age \< 12 years * DONOR: Identical twin * DONOR: Pregnancy * DONOR: Infection with HIV * DONOR: Inability to achieve adequate venous access * DONOR: Known allergy to filgrastim (G-CSF) * DONOR: Current serious systemic illness