Not khan the lord or prince, but khan a building for the accomodation of travellers, as in gymkhana "place of public resort at a station, where the needful facilities for athletics and games of sorts are provided." According to the OED, gymkhana is a refashioning of Hindi gend-khāna "ball house", from गेंद gend "ball" plus ख़ाना ḵẖānā "house", borrowed from the Persian word below.
Proto-Indo-European *h₁ēs- "to sit" (a lengthened-grade form of *h₁es- "to be") in the suffixed form *h₁ēs-en-o- became Iranian *āhanam "seat" then Middle Persian خان ḵẖān "house".
In Sanskrit *h₁ēs- became ās "to sit" and आसन āsana "sitting", used as a term for yoga positions.
khan meaning "lord" or "prince", and probably also the source of Shahrukh Khan's name, is from Turkish.