T Cell Receptor Gene Therapy Targeting KK-LC-1 for Gastric, Breast, Cervical, Lung and Other KK-LC-1 Positive Epithelial Cancers

* INCLUSION CRITERIA: * Participants must have histologically or cytologically confirmed Kita-Kyushu Lung Cancer Antigen-1 (KK-LC-1) positive epithelial cancer (KK-LC-1 positivity assay performed at Rutgers Cancer Institute of New Jersey). KK-LC-1 expression will be determined by immunohistochemistry (IHC). KK-LC- 1 score of 25% or greater will be considered positive. * Participants must be HLA-A 01 (human leukocyte antigen serotype within HLA-A "A" serotype group) by low resolution typing, and HLA-A 01:01 by one of the high-resolution type results * Measurable metastatic or refractory/recurrent KK-LC-1+ epithelial cancer (determined by Immunohistochemistry (IHC). * All participants must have received prior first line standard therapy or be ineligible to receive available therapies with known survival benefit. * Participants with three or fewer brain metastases that have been treated with surgery or stereotactic radiosurgery are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month before protocol treatment. Participants with surgically resected brain metastases are eligible. * Age \>18 years of age. Because no dosing or adverse event data are currently available on the use of KK-LC-1 T cell receptor (TCR) in participants \<18 years of age, children are excluded from this study. * Eastern Cooperative Oncology Group (ECOG) performance status \<1. * Participants must have adequate organ and marrow function as defined below: * leukocytes \>=3,000/mcL * absolute neutrophil count \>=1,500/mcL * platelets \>=100,000/mcL * hemoglobin \>=9.0 g/dL * total bilirubin within normal institutional limits * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT)/alanine aminotransferase (ALT)/serum glutamic-pyruvic transaminase (SGPT) =\<2.5 X institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance \>=60 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal. -Serology: * Seronegative for human immunodeficiency virus (HIV) antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who are HIV seropositive can have decreased immune-competence and thus are less responsive to the experimental treatment and more susceptible to its toxicities.) * Seronegative for hepatitis B antigen, and seronegative for hepatitis C antibody. If hepatitis C antibody test is positive, then the participant must be tested for the presence of antigen by reverse transcription-polymerase chain reaction (RT-PCR) and be hepatitis C virus ribonucleic acid (HCV RNA) negative. * The effects of the study agent on the developing human fetus are unknown. For this reason and because other therapeutic agents used in this trial are known to be teratogenic, individuals of child-bearing potential and their partners must agree to use adequate contraception (barrier method of birth control; abstinence) prior to study entry and up to twelve (12) months after treatment. Individuals of childbearing potential must have a negative pregnancy test. Individuals of childbearing potential are defined as all individuals except individuals who are postmenopausal or who have had a hysterectomy. Postmenopausal will be defined as individuals over the age of 55 who have not had a menstrual period in at least one year or a participant that has received a treatment (i.e., chemotherapy or pelvic radiation) that has resulted in them becoming postmenopausal. Should an individual become pregnant or suspect pregnancy while individuals or partner is participating in this study, individuals should inform treating physician immediately. * Because there is a potential risk for adverse events in nursing infant's secondary to treatment of the mother with KK-LC-1 TCR transduced peripheral blood lymphocytes (PBLs), breastfeeding should be discontinued if the mother is treated with KK-LC-1 TCR transduced PBL. These potential risks may also apply to other agents used in this study. * More than four weeks must have elapsed since any prior systemic therapy at the time the participant receives the KK-LC-1 TCR T cells. * Participants may have undergone minor surgical procedures within the past three weeks, as long as all toxicities have recovered to Grade 1 or less * Ability of subject to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Participants who are receiving any other investigational agents. * History of severe immediate hypersensitivity reaction to cyclophosphamide, fludarabine or aldesleukin. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. Participants with abnormal pulmonary function tests but stable obstructive or restrictive pulmonary disease may be eligible. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). * Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who have decreased immune competence may be less responsive to the experimental treatment and more susceptible to its toxicities). * Participants with autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis or pancreatitis, and systemic lupus rythematosus. Hypothyroidism, vitiligo and other minor autoimmune disorders are not exclusionary. * Participants on active systemic immunosuppressive therapy that cannot be safely withheld. * Participants with a history of coronary revascularization or ischemic symptoms unless participant has a normal cardiac stress test. * Any condition which would, in the opinion of the Principal Investigator, indicate that the subject is a poor candidate for the clinical trial or would jeopardize the subject or the integrity of the data obtained. * Documented left ventricular ejection fraction (LVEF) \<= 45%