It's difficult to advise without knowing your means of transport between Florence and Istanbul, and you only have a week to spare. You tagged this post for Greece and Turkey, which tells me you are interested in a stop or stops in Greece before making your way to Istanbul, so you could fly to Athens, then go on by air from there to Istanbul.

Having already been to Athens and "a few Greek islands" you may have an idea of where you would like to return for an in-depth visit. Presumably one of those Greek islands you have already visited is Patmos. You may want to consider the monasteries at Meteora, or the island of Tinos, which is a place of pilgrimage for Orthodox Greeks. If you haven't already been to the Christian and Byzantine Museum in Athens, I would suggest that as well.

Another idea would be the Byzantine city of Thessaloniki, one of the most important cities of the Byzantine Imperium.
From there you can do some very interesting day trips to Veria, Kavala, Dion, Philippi and Amphipolis.

Thank you both.

Yes, we are most interested in going to Greece and Turkey. We haven't yet thought very much about, let alone decided on our means of transport. We had planned to decide that based on where we wanted to go. I had vaguely considered a flight from Florence to Athens or Thessaloniki, ferrying to one Greek island, and then flying to Istanbul for the last few days.

At this point we are more looking for suggestions of places to visit, and then we will plan our transport based on those.

Thanks again!