Panobinostat (LBH589) and Imatinib Mesylate in Treating Patients With Previously Treated Chronic Phase Chronic Myelogenous Leukemia

Inclusion Criteria: * Ability to provide written informed consent obtained prior to participation in the study and any related procedures being performed * CML CP patients who have been treated with and tolerated Imatinib for 1 year or more, have achieved at least major cytogenetic response and continue to be BCR-ABL positive (Patients should be receiving Imatinib at a dose of 400 daily at the time of entry into the study) * ANC and PLT need to be in the normal range * Serum albumin \>= 3g/dL * AST/SGOT and ALT/SGPT =\< 2.5 x upper limit of normal (ULN) * Serum bilirubin =\< 1.5 x ULN * Serum creatinine =\< 1.5 x ULN or 24-hour creatinine clearance \>= 50 ml/min * Serum potassium \>= lower limit of normal (LLN) * Serum phosphorus \>= LLN * Serum total calcium (corrected for serum albumin) or serum ionized calcium \>= LLN * Serum magnesium \>= LLN * ECOG performance status of =\< 2 Exclusion Criteria: * Prior treatment with an HDAC inhibitor * Patient who have been treated with Imatinib \< 1 year or patients are currently being treated with Imatinib at a dose \> 400 mg daily * Impaired cardiac function including any one of the following: Screening ECG with a QTc \> 450 msec; Patients with congenital long QT syndrome; History or presence of sustained ventricular tachycardia; Any history of ventricular fibrillation or torsades de pointes; Bradycardia defined as heart rate \< 50 beats per minute (patients with a pacemaker and heart rate \>= 50 beats per minute are eligible); Patients with a myocardial infarction or unstable angina within 6 months of study entry; Congestive heart failure (NY Heart Association class III or IV); Right bundle branch block and left anterior hemiblock (bifascicular block) * Uncontrolled hypertension * Concomitant use of drugs with a risk of prolonging the QT interval or inducing torsades de pointes * Concomitant use of CYP3A4 inhibitors * Patients with unresolved diarrhea \> CTCAE grade 1 * Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of oral LBH589 * Other concurrent severe and/or uncontrolled medical conditions * Patients who have received chemotherapy, any investigational drug or undergone major surgery \< 4 weeks prior to starting study drug or who have not recovered from side effects of such therapy * Concomitant use of any other anti-cancer therapy or radiation therapy * Patients being treated with Coumadin (unless patients who require anticoagulation can be switched to a low-molecular weight or standard heparin) * Female patients who are pregnant or breast feeding or patients of reproductive potential not willing to use a double method of contraception including a barrier method (i.e. condom) during the study and 3 months after the end of treatment (Women of childbearing potential \[WOCBP\] must have a negative serum pregnancy test within 7 days of the first administration of oral LBH589) * Male patients whose sexual partners are WOCBP not willing to use a double method of contraception including condom during the study and 3 months after the end of treatment * Patients with a history of another primary malignancy within 5 years other than curatively treated CIS of the cervix, or basal or squamous cell carcinoma of the skin * Patients with known positivity for human immunodeficiency virus (HIV) or hepatitis C; baseline testing for HIV and hepatitis C is not required * Patients with any significant history of non-compliance to medical regimens or with inability to grant a reliable informed consent