We had a brilliant stay at Plito's in Amorgos. The flat is located a couple of minutes from some lovely bars / restaurants but is still very peaceful with stunning views. The flat is well made up and clean, with everything we needed.
I would generally recommend Katapola as a place to stay if travelling to Amorgos too. It's quaint, where the ferry arrives and other places in Amorgos are easily accessible by car.