ADCT-602 in Treating Patients With Recurrent or Refractory B-cell Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients with relapsed or refractory B-ALL. Philadelphia chromosome positive (Ph+) ALL is allowed after failing either first or second generation tyrosine kinase inhibitor. Note: Patients in first relapse with complete remission (CR1) duration \> 12 months are excluded * Expression of CD22 in \>= 20% blasts (assessed by flow-cytometry or immunohistochemistry) * Marrow blast count \>= 5% * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Serum creatinine =\< 1.5 mg/dL. If the patient has a creatinine \> 1.5 mg/dL, creatinine clearance must be \> 60 mL/min/1.73 m\^2, as calculated by the Cockcroft and Gault equation, or modification of diet in renal disease (MDRD) formula or 24-hour urine analysis * Serum alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2 times the upper limit of normal (ULN); =\< 5 times ULN if there is liver or bone involvement * Total bilirubin =\< 1.5 times ULN. Patients with known Gilbert's syndrome may have a total bilirubin up to =\< 3 times ULN. * NOTE: In patients (pts) with elevated total bilirubin due to increased indirect bilirubin, patients with direct bilirubin =\< 1.5 x ULN are eligible * Left ventricular ejection fraction (LVEF) \>= 45% * Negative urine or serum beta-human chorionic gonadotropin (B-HCG) pregnancy test within 7 days prior to the cycle 1, day 1 visit, for women of child-bearing potential. Women of child bearing potential must agree to use an effective method of con…