If you want to have the real Kimolos experience and feel like a local, that is the place to stay - literally inside the medieval castle, next to the ancient agora and with two small balconies, one in the front and one in the back, with views to beautiful spots of the village. The communication with Fileni was great since the beginning and she assisted a lot when we were organizing our trip.
The whole island gives the feeling of something authentic, untouched and from another time. One of the best places i have visited in Greece, and i am Greek.
Just be prepared for some walking because cars are not aloud inside the village.