Panobinostat With Fludarabine and Cytarabine for Treatment of Children With Acute Myeloid Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * Participants must have a diagnosis of AML or MDS and must have disease that has relapsed or is refractory to chemotherapy, or that has relapsed after hematopoietic stem cell transplantation (HSCT). * Refractory disease is defined as persistent disease after at least two courses of induction chemotherapy. * Patients with AML must have ≥ 5% leukemic blasts in the bone marrow or increasing levels of minimal residual disease (MRD) in the bone marrow as assessed by flow cytometry. If an adequate bone marrow sample cannot be obtained, patients may be enrolled if there is unequivocal evidence of leukemia in the peripheral blood. * Adequate organ function defined as the following: * Direct bilirubin ≤ 1.5 x institutional upper limit of normal (IULN) * AST (SGOT) and ALT (SGPT) ≤ 2.5 x ULN * Creatinine ≤ 1.5 x ULN for age * Serum albumin \> 3.0 g/dl * Left ventricular ejection fraction ≥ 40% or shortening fraction ≥ 25%. * Age ≤ 24 years * Patients must be able to swallow capsules * Performance status: Lansky ≥ 50 for patients who are ≤ 16 years old and Karnofsky ≥ 50% for patients who are \> 16 years old. * Patients must have fully recovered from the acute effects of all prior therapy and must meet the following criteria: * At least 14 days must have elapsed since the completion of myelosuppressive therapy * At least 24 hours must have elapsed since the completion of low-dose chemotherapy, such as hydroxyurea or low-dose cytarabine (up to 200 mg/m\^2/day). * For patients who have received prior HSCT, there can be no evidence of GVHD and greater than 60 days must have elapsed since the HSCT. Patients cannot be receiving therapy, including steroids, for the treatment or prevention of GVHD. All such medications must be discontinued at least 24 hours prior to enrollment. * Body Surface Area: Because the smallest capsule size available for the panobinostat is 10 mg, the minimum BSA allowed for enrollment at Dose Level 1 to 0.85 m\^2. The minimum for Dose Level 2 is BSA=0.6 m\^2 and the minimum for Dose Level 3 is BSA=0.42 m\^2. Exclusion Criteria: * Must not be pregnant or breastfeeding. Female patients who are sexually active and of child-bearing potential must agree to use dual methods of contraception and have a negative serum pregnancy test at screening, and male patients who are sexually active must use an effective barrier method of contraception if sexually active with a female of child-bearing potential. For both male and female patients who are sexually active, effective methods of contraception must be used throughout the study and for three months following the last dose. Abstinence is an acceptable form of contraception. * Patients with Down syndrome, acute promyelocytic leukemia, juvenile myelomonocytic leukemia, Fanconi anemia, Kostmann syndrome, Shwachman syndrome, or other bone marrow failure syndromes are not eligible. * Use of investigational agents within 30 days. * Any significant concurrent disease, illness, or psychiatric disorder that would compromise patient safety or compliance, study participation, follow up, or interpretation of study research. * Uncontrolled infection within one week of the first dose. Infections controlled on concurrent anti-microbial agents are acceptable, and anti-microbial prophylaxis per institutional guidelines are acceptable. * Known human immunodeficiency virus infection (pre-study testing not required). * Patient with diarrhea \> CTCAE grade 2. (CTCAE version 4.0) * Impaired cardiac function or clinically significant cardiac diseases, history of arrhythmia (including ventricular fibrillation or torsade de pointes), bradycardia \<50 bpm, screening ECG with prolonged QTc (\> 450 msec), uncontrolled hypertension or any history or presence of sustained ventricular tachyarrhythmia. * Impairment of GI function or GI disease that may significantly alter the absorption of panobinostat. * Patients using medications that have a relative risk of prolonging the QT interval or inducing torsade de pointes if treatment cannot be discontinued or switched to a different medication prior to starting treatment. Granisetron may be administered, but antiemetics associated with QT prolongation (e.g., ondansetron) are not allowed.