I'm looking for something written by the English traveling to Constantinople, or preferably by Turks traveling to England if available. I've already looked at Richard Hakluyt's The Principal Navigations, Voyages, Traffiques, and Discoveries of The English Nation, and am wondering if there was something else, perhaps something translated from Turkish?
Sure.
Henry Blount, A Voyage into the Levant (1636), and Thomas Dallam's diary (c1600) are available for free online.
The prologue of Gerald MacLean, The Rise of Oriental Travel: English Visitors to the Ottoman Empire, 1580-1720, translates an account of the Ottoman side of part of Dallam's visit.