I think 6 days in Istanbul is the perfect duration to be able to see the most of it, especially the Islamic heritage of the Ottoman Empire.
How much things you can see per day will depend mainly on how active you are and if you will be accompanied by any kids.
If you will be traveling alone on with some adult people, I really think you can see much more things in a single day.
Example:
Day 1 -
Blue Mosque & Hagia Sophia are close to each other and can be seen in 4 hours, leaving you at least 6 hours of activity. You can also explore the Sultan Ahmet Square, Arasta bazaar and the small and old Ottoman alleys around the SultanAhmet area
Turkish and Islamic Arts Museum
Day 2
Eyup Mosque is very far away and I strongly advise to start your day itinerary by visiting it early morning (8 am), so it would not waste the rest of the day for you
Day 3
Princess islands tour full day - Personally I don’t think this is worth is for your first time Trip to Istanbul, as there are other much interesting things to see. Personally I believe the Full Bosphoros Cruise is much more interesting.
For accommodation, I think a hotel around Taksim/Istiklal street will be the best choice, as it will allow you to enjoy the nice atmosphere, coffee shops and restaurants of Istiklal street by night.
Here is an itinerary and guide I have created personally based on my previous trips to Istanbul which can give you some good ideas on how to schedule your days to see the max number of historical attractions. It also has a lot of links to other blogs that give detailed information about stuff you might want to know
https://tripblan.com/trip/the-ultimate-one-week-istanbul-itinerary-and-guide
All the best