Moxetumomab Pasudotox-tdfk (Lumoxiti(TM)) and Either Rituximab (Rituxan(R)) or Ruxience for Relapsed Hairy Cell Leukemia

* INCLUSION CRITERIA: * Diagnosis of hairy cell leukemia (HCL) or Hairy cell leukemia variant (HCLv). * Treatment required for either 1) Absolute neutrophil count (ANC) \<1/nL, 2) Hemoglobin \<10g/dL, 3) Platelets\<100/nL, 4) symptomatic splenomegaly, or 5) enlarging HCL mass \> 2cm in short axis. Patients who have eligible blood counts within 4 weeks from the initiation of study will not be considered ineligible if subsequent blood counts prior to enrollment fluctuate and become ineligible up until the time of enrollment. * Patients must be Pseudomonas-immunotoxin naive. * HCL or HCLv with at least 1 prior purine analog, and, for HCL patients with \>=2-years 1 month response, at least 1 other therapy. Age greater than or equal to 18 years as the disease under study, HCL/HCLv, has not been reported in children \< age 18. * Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to2 (Karnofsky greater than or equal to 60%) * Patients must have adequate organ and marrow function as defined below: * Total bilirubin less than or equal to 1.5 mg/dL, unless consistent with Gilbert's (ratio between total and direct bilirubin \> 5) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3x upper limit of normal (ULN) * Alkaline phosphatase \< 2.5 ULN * Serum creatinine less than or equal to 1.5 mg/dL or creatinine clearance greater than or equal to 60 mL/min by Cockcroft-Gault equation, where creatinine clearance = (140-age)(Kg weight)/(72 x Creatinine) * Serum albumin greater than or equal to 2 g/dL * Partial thromboplastin time (PTT) or Prothrombin time (PT)/International Normalized Ratio \< 2.5x ULN (If on warfarin, PT/INR \< 3.5x ULN; If on any other anticoagulation, Prothrombin time (PT) \< 2.5x baseline) * Fibrinogen greater than or equal to 0.5 lower limit of normal * The effects of moxetumomab pasudotox-tdfk and rituximab/Ruxience on the developing human fetus are unknown therefore participants must use effective methods of contraception as directed below. * Females of childbearing potential (\< 50 years) who are sexually active with a non-sterilized male partner must use a highly effective method of contraception prior to study entry and or the duration of study participation and must agree to continue using such precautions for 12 months after completion of rituximab/Ruxience administration. Contraception after this point should be discussed with a responsible physician. Periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of contraception. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or those who are premenarchal or postmenopausal (defined as 12 months with no menses without an alternative medical cause). A highly effective method of contraception is defined as one that results in a low failure rate (i.e., less than 1% per year) when used consistently and correctly. Not all methods of contraception are highly effective. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Non-sterilized males who are sexually active with a female partner of childbearing potential must use an effective method of contraception from Day 1 until 90 days after receipt of the final dose of investigational product. It is required that a female partner of a male subject also use an effective method of contraception throughout this period. * Ability of subject to understand and the willingness to sign a written informed consent document. * Patients must be willing to co-enroll in the investigators companion protocol 10C-0066 titled Collection of Human Samples to Study Hairy Cell and other Leukemias, and to Develop Recombinant Immunotoxins for Cancer Treatment. EXCLUSION CRITIERIA: * Patients who have had chemotherapy, immunotherapy or radiotherapy within 4 weeks or treatment with rituximab/Ruxience within last 3 months prior to initiation of treatment. * Patients who are receiving any other investigational agents. * Breastfeeding within the projected duration of the study, starting with the screening visit through 6 months after the last dose of rituximab/Ruxience. Pregnant women are excluded from this study because moxetumomab pasudotox-tdfk and rituximab/Ruxience are agents with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with moxetumomab pasudotox-tdfk and rituximab/Ruxience, breastfeeding should be discontinued if the mother is treated with moxetumomab pasudotox-tdfk and rituximab/Ruxience. * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, uncontrolled hypertension, uncontrolled pulmonary infection, pulmonary edema or psychiatric illness/social situations that would limit compliance with study requirements. * Patients with retinal or choroidal detachment. * Positive for Hepatitis B core antibody or surface antigen unless the patient is on Tenofovir or Entecavir and Hepatitis B Viral deoxyribonucleic acid (DNA) load is \<2000 IU/mL * Active second malignancy requiring treatment other than minor resection of indolent cancers like basal cell and squamous skin cancers. * Human immunodeficiency virus (HIV)-positive patients unless taking appropriate anti-HIV medications with a cluster of differentiation 4 (CD4) count of \> 200. Otherwise, there may be an increased risk of lethal infections when temporarily suppressing normal B-cells. * History of an allogeneic bone marrow transplant. * Patients with a history of both thromboembolism and known congenital hypercoagulable conditions. * Radioimmunotherapy within 2 years prior to enrollment in the study. * Patients with history of thrombotic microangiopathy or thrombotic microangiopathy/hemolytic uremic syndrome (HUS). * Patients with corrected QT interval (Frederica) elevation \> 500 msec (manually over-read by medically qualified person) based on at least two separate 12-lead ECGs. * Patients on high dose estrogen (defined as \> 0.625 mg/day of an estrogen compound). * Oxygen saturation at rest \< 88% measured by pulse oximetry or partial pressure of oxygen (PaO2) less than or equal to 55 mm Hg. * Patients with life expectancy of less than 6 months. * Patients with clinical evidence of disseminated intravascular coagulation (Grade 3-4). * Patients with \< 50% of predicted forced expiratory volume (FEV1) or \< 50% of predicted diffusing capacity for carbon monoxide, corrected for hemoglobin concentration and alveolar volume (DLCO). Note: Patients with no prior history of pulmonary illness are not required to have pulmonary function testing (PFT). Forced expiratory volume will be assessed after bronchodilator therapy.