When Katherine has an allergic reaction to an ingredient in her meal, likely in the hot sauce, Jack diagnoses her & recommended a course of treatment, including medication. This plot point could have worked had the character been an NP (nurse practitioner), but, as an RN, he was not only not licensed or qualified to do those things, but still should have known to take no chances with a food allergy & taken her to an ER.
During a game of chess, the black king is moved more than one square (the maximum allowed for a king) along a diagonal like a queen or bishop.
Jack certainly would google Katherine after visiting her office.