We loved staying in Burcu's apartment - the space is clean, spacious, and in a terrific location near the Tophane tram stop. Would highly recommend staying here over a more touristy neighborhood like Sultanahmet. Burcu was very easy to communicate with, and even helped us with an early check-in, despite the fact that the apartment was scheduled to be cleaned between guests at the time when we needed to drop our luggage off.
Two things to note - in the summer the apartment gets stuffy, and there is only one fan. The apartment is also located up a steep hill (and then three more flights of stairs). I'm pretty reasonably fit and in my 20s and was still exhausted by the time we'd get home (granted we also chose a hot time of year to visit). Something to think about, in addition to the fact that taxis are sparse until you get to the main road (though the main road is not far).
Overall - excellent neighborhood and host. Definitely recommend as home base for exploring Istanbul.