This clinical trial studies fludarabine phosphate, low-dose total-body irradiation, and peripheral blood stem cell transplant followed by donor lymphocyte infusion in treating older patients with chronic myeloid leukemia. Giving chemotherapy and total-body irradiation before a donor bone marrow transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil after the transplant may stop this from happening. Once the donated stem cells begin working, the patient's immune system may see the remaining cancer cells as not belonging in the patient's body and destroy them (called graft-versus-tumor effect). Giving an infusion of the donor's white blood cells (donor lymphocyte infusion) may boost this effect.