Does an auto immune disorder have anything to do with a defect in killer t cells?

Maybe, maybe not: There are many different types of autoimmune disorders, many of which are not associated with immune deficiencies. Certain immune deficiencies (e.g., wiskott-aldrich syndrome, cvid, apeced) can be associated with autoimmunity, but only some are associated with defects in "killer" t cells. Some disorders of immune regulation (e.g. Hlh) can act like autoimmunity. See an immunologist for more info.