Well, I would say that if you want to stay in Istanbul, stay in THIS neighborhood. Did a couple days elsewhere (Taksim) and walked around quite a bit, so I can say that we felt relieved when getting back around Sultanahmet, with less traffic and paced down atmosphere. It is indeed still in the touristic area (lots of small hotels around), really close to the site you want to visit (go to Hagia Sofia BEFORE 8 AM), but at the same time, the shops there feel more "local", are friendly and not overpriced ( actually they were amongst the cheapest we found).
Hikmet was a great host, and arranged for parking. This might be the single problem: to get there with a car can be challenging to say the least, really, if you get there at rush hour, just go up Şht. Çeşmesi Sokağı (even if there is a car blocking the end or going down) and meet with Hikmet, DO NOT try to drive around, you will loose hours. Then leave your car and just do everything by foot or other transports, I know that I did not want to drive around until we left.
Enjoy the terrace at the top of the building, there's even a washing machine, and the view to the boats is great. The mosque just next door will wake you up, but really, after 2-3 days, we would not mind anymore. Apartment was clean, spacious.
I recommend also the "Turkish cuisine" restaurant, I know, really "tourist trap" name (and it can be a bit slow), but actually the food is great, the service good, and at relatively low price (went for diner almost every day there). The rabbit hole is always a great place to get your delicious breakfast. If I were to come back to Istanbul, I would undoubtedly lodge there.