Mesothelin-Targeted Immunotoxin LMB-100 in Combination With Tofacitinib in Persons With Previously Treated Pancreatic Adenocarcinoma, Cholangiocarcinoma and Other Mesothelin Expressing Solid Tumors

* INCLUSION CRITERIA: * Patients must have histologically confirmed solid tumor malignancy for which no curative therapy exists. * Pancreatic adenocarcinoma, extrahepatic cholangiocarcinoma or epithelioid subtype of mesothelioma, as determined by National Cancer Institute (NCI) Laboratory of Pathology, OR for all other tumor types, at least 20% of tumor cells must express mesothelin. Determination can be made using archival tumor tissue or fresh biopsy if archival tumor tissue is not available. * All patients must have evaluable disease (i.e., measurable per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. or by following carbohydrate antigen 19-9 (CA19-9) tumor marker). Patients in the expansion cohort must have measurable disease, per RECIST 1.1. evaluation of measurable disease. * Patients must have received at least one prior standard systemic treatment regimen for advanced disease OR be ineligible to receive available standards due to co-morbidities, prior toxicity, lack of standard options for tumor type, or having received all standards available for prior treatment of early-stage disease OR have refused first-line standard systemic treatment but have received prior anti-cancer treatments. * Patients with deficient Mismatch Repair (dMMR)/high levels of MicroSatellite Instability (MSI-H) disease must have received at least one prior anti-programmed cell death 1 (PD1) therapy, be ineligible to receive this treatment due to concurrent medical conditions or have refused this therapy. * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-2 * Age \>=18 years. Because no dosing or adverse event data are currently available on the use of LMB-100 alone or in combination with tofacitinib in persons with \<18 years of age, children are excluded from this study. * Patients must be more than 14 days removed from most recent minor surgical procedure (such as biliary stenting), 28 days from most recent major surgical procedure and 14 days from radiation therapy, systemic treatments (such as chemotherapy), or experimental drug treatment. All acute toxicities from prior treatment must have resolved to grade 1 or less except alopecia, anemia, peripheral neuropathy, or endocrinopathies corrected by replacement therapy. * Adequate hematological function: neutrophil count of \>= 1.5 x 10\^3 cells/micro liters, platelet count of \>= 85,000/micro liters, hemoglobin greater than or equal to 9 g/dL * Serum albumin \>= 2.5 mg/dL without intravenous supplementation * Adequate liver function: Bilirubin \<2.5 x upper limit of normal (ULN) for all, aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 3 x ULN except for patients with significant tumor burden in their liver where AST and ALT \< 5x ULN is acceptable in the absence of other etiologies for transaminitis * Adequate renal function: creatinine clearance \[Estimating glomerular filtration rate (EGFR) method or measured\] \>= 50 mL/min. Measured clearance will be used if both numbers are available. * Must have left ventricular ejection fraction \>= 50% * Must have an ambulatory oxygen saturation of \> 88% on room air * The expansion phase patients must meet all eligibility criteria above AND must have diagnosis of pancreatic adenocarcinoma or extrahepatic cholangiocarcinoma with pathology confirmed to be consistent with one of these diagnoses by NCI Laboratory of Pathology. * The effects of LMB-100 alone or in combination with tofacitinib on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry until 3 months the last dose of study therapy. 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. * Ability of participant to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Known or clinically suspected central nervous system (CNS) primary tumors or metastases including leptomeningeal metastases as CNS penetration of LMB-100 is expected to be poor. CNS metastases are permitted if they have been previously treated, are asymptomatic, and have had no requirement for steroids or enzyme-inducing anticonvulsants in the last 14 days. * Evidence of significant, uncontrolled concomitant diseases which could affect compliance with the protocol or interpretation of results, including significant pulmonary disease other than that related to the primary cancer, uncontrolled diabetes mellitus, and/or significant cardiovascular disease (such as New York Heart Association Class III or IV cardiac disease, myocardial infarction within the last 6 months, unstable arrhythmias, unstable angina, or clinically significant pericardial effusion). * Any known diagnoses, metabolic dysfunction, physical examination finding, or clinical laboratory finding giving reasonable suspicion of a disease or condition (other than mesothelin \[+\] cancer diagnosis) that would contraindicate the use of an investigational drug, interfere with tumor measurement or lead to a life expectancy of less than 6 months as judged by the investigator. * Contraindication to receiving prophylactic doses of low-molecular weight heparin (LMWH) or direct oral anticoagulants (DOAC) such as current active bleeding (except for grade 1 hematuria or epistaxis), recent history of significant bleeding without subsequent effective medical or surgical intervention, known history of gastric varices, uncontrolled malignant hypertension, history of coagulopathy that confers increased risk of bleeding. Patients on concurrent treatment with anti-platelet agents such as aspirin or clopidogrel are eligible if deemed to have acceptable risk of bleeding in consultation with Hematologist. Patients already receiving prophylactic or therapeutic doses of anticoagulant (heparin-based, or DOAC) for at least 4 weeks with no indication of significant bleeding while on therapy are considered NOT to have a contraindication to this therapy. * Inability to administer or unwillingness to comply with recommended venous thromboembolism (VTE) prophylaxis for the duration of study treatment. * Prior diagnosis of hematologic malignancy * Active or uncontrolled infections (including tuberculosis, human immunodeficiency virus (HIV), hepatitis B virus (HBV), or hepatitis C virus (HCV) or reasonable clinical suspicion of an active infection (such as cholangitis) as tofacitinib suppresses lymphocyte signaling and will impair host response to infection * Latent tuberculosis (TB) infection as identified by interferon-gamma release assay (IGRA). If IGRA is indeterminate, tuberculin skin test (TST) may be used to determine status. * Live attenuated vaccinations within 14 days prior to treatment. * Use of a strong inhibitor or inducer of cytochrome P450 3A4 (CYP3A4) within 14 days prior to enrollment or similarly updated source for a list of such agents) * Inability to take or digest oral medication. * Dementia or altered mental status that would prohibit informed consent. * Pregnant women are excluded from this study because the effects of LMB-100 and/or tofacitinib on the developing fetus are unknown and may have the potential to cause 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 LMB-100 and/or tofacitinib, breastfeeding should be discontinued if the mother is treated with either of these agents. * Baseline corrected QT interval by Fredericia (QTcF) interval of \> 470 ms, participants with baseline resting bradycardia \< 45 beats per minute, or baseline resting tachycardia \>100 beats per minute. * Participants with contra-indication and/or history of severe hypersensitivity reactions to any components related to LMB-100 and tofacitinib. * Patients who have previously received LMB-100 (and therefore have high-levels of preexisting anti-drug antibodies (ADAs) to drug)