This study will evaluate the effectiveness of an experimental drug called LMB-2 for treating chronic lymphocytic leukemia (CLL) in patients who have a protein called cluster of differentiation 25 (CD25) on their cancer cells. LMB-2 is a recombinant immunotoxin. It is made up of two parts: a genetically engineered monoclonal antibody that binds to cancer cells with CD25 on their surface, and a toxin produced by bacteria that kills the cancer cells to which it binds. LMB-2 has killed CD 25-containing cells in laboratory experiments and has caused tumors in mice to shrink. Preliminary studies in humans have shown some effectiveness in shrinking tumors in patients with various types of lymph and blood cancers.