This study will examine the use of the humanized Mik-beta-1 (Hu-Mik-beta1) antibody in patients with T-cell large granular lymphocytic leukemia (T-LGL). Patients with T-LGL often have reduced white blood cells, red blood cells, and platelets, and increased numbers of abnormal cells called large granular lymphocytes (LGLs). Patients may have recurrent infections, anemia, or abnormal bleeding. Hu-Mik-beta1 attaches to LGL cells and blocks the action of growth factors called interleukins that stimulate LGL growth. Blocking these interleukins may stop T-LGL leukemia cells from growing. This study will determine the dose and frequency of treatment with Hu-Mik-(SqrRoot) 1 that can safely be given to patients to coat the surface of their leukemic cells with antibody, determine how long the antibody lasts in the blood after injection, and examine the side effects and possible benefits of the drug in these patients.