If you go in the springtime, it is worth walking through Gulhane Park, which is in the shadow of Topkapi palace. Very relaxing place amid the hustle and bustle with wonderful flower beds and there is also a museum in the grounds showing scientific inventions.
A visit to Pera Palace hotel would be good - newly refurbished. I have not seen since refurbishment but there used to be a cafe, very art deco, which I would have thought would have been kept. Also this is the hotel where Agatha Christie wrote Murder on the Orient Express! Hotel is situated near to Istiklal Caddesi, and you can reach this by using the Tunel - worth seeing.
This part of Istanbul has different architecture to Sultanahmet, churches and embassy buildings that are interesting though. Also Cicek Pasagi off Istiklal Cad.