I do recommand Villa Lydie.
The villa was big enough to accomodate all the group of 10 (parents, teenagers and children) so we were able to enjoy moments together and privacy. It was comfortable. You have small bathrooms in 3 rooms plus only one with a big shower. Even if there is no AC the villa is fresh enough if you close the windows during the day when there is a lot of sun. The swimming pool is small but sufficient to refresh and enjoy the beautifull views on Chora. You need a car if you stay there. Beware as you may have lots of wind. It is pleasant as it refresh the hot weather but also might sometimes be disturbing.
Ioannis was very caring. Whenever we experience some issues and contact him, he was very responsive. Thanks!