The Istanbul is almost immediately opposite the entrance to the lower church. Its got a lovely open upstairs seating area, and Engin is very much the host with the most, and if theres lots of people will turn it into a party atmosphere and get people dancing.(evenings)
If you go in the daytime take swimming stuff as you will be able to use the pool.
Same next door at Muzzys Place, you can use the pool even if you only stop for a drink.
Cin Bal is very good, theres a good website telling about it's origins, Kebap Evi is similar with you being able to cook at your own barbeque, its a great experience, although a bit scary as everyone else is mostly Turkish and know what to order and what to do and us Brits stand out as a bit vague!
Also, if you go further down the village and turn right at the 'clocktower' keep walking, past Kayapark and ahead there is a small garden restaurant on a corner, can't remember what it was called, but fantastic food, and excellent value, and the pop up tooth pick dispensers keep the kids amused for hours ( I searched for weeks to find one on the net for my son afterwards!)
Anywhere in Kaya is fantastic, enjoy.