In April 2010, I stayed for ten days at the Deniz Konak (http://www.denizhouses.com/). Clean, comfortable, friendly English-speaking staff, good (but not great) breakfasts, and an ideal location for exploring Sultanahmet by foot. A pleasant five-minute walk from the Blue Mosque / Hagia Sophia / Topkapi. It's a converted 19th-Century Ottoman house in a neighborhood that is tourist-oriented--thus lots of options for food and drink--but still retains a very local feel, so I think this matches your desire for exotic charm. The dining area of the hotel is on the top floor, which has small outside patios atop the building providing views toward both the Bosporous and the city in the other direction. Neither view is particularly stunning in a postcard-like way, but you get a strong feel for the city as well as a relatively quiet place for peaceful respite, especially during the evening call to prayer. The first half of this video was taken from my room: http://vimeo.com/15964020
I would definitely stay there again, although I think there are probably several comparable alternatives in the vicinity. If it's important to you, the Deniz Konak does have free Wi-Fi, although the signal isn't strong and you might find it necessary to sit in their cozy reception area to get it to work. They are affiliated with Backpackers Travel, and thus well connected to set you up with tours, ground transportation, etc., should you desire. Here's more info on them: http://www.turkeytravelplanner.com/tourop/bakpak/.
Have a great trip!