When visiting Istanbul, staying in the Old City probably seems like the best (at least most convenient) option. I'd actually recommend venturing outside of Sultanahmet.
For the first part of our trip, we stayed in Cihangir at the Witt Istanbul Hotel. Cool, modern, and an easy commute to the Old City via the tram. The neighborhood is funky and filled w/ cafes, which was a nice break from the crowds across the Horn.
After touring other parts, we returned to Istanbul for our "splurge" hotel. This time we stayed in Çengelköy on the Asian side at the Sumahan on the Water. No, it wasn't cheap, but it did provide stunning views of the Bosphorus Bridge and an opportunity to experience another (less touristic) neighborhood.
This is the view from our room at the Sumahan. The room has floor to ceiling windows. And since the bed is angled for optimal views, you can lay in bed and watch the sunset.