We (two people) stayed 6 nights at the Persephone and enjoyed it very much. The place is perfectly located with a small terrasse overlooking the valley to the west. Using binoculars you can see the ancient "Lion of Kea", just a 15 minutes walk from Ioulis.
The appartment is very clean and Niki, the owner, responses quickly. The large bed in the separate sleeping room and the bench in the kitchen/living room provide comfortable living and sleeping arrangement, even if you want to sleep separately. Due to the lack of hooks and shelfs we rearranged and stored and most things, such as decoration, ironing and cleaning equipment, in the generous storage room underneath the benches. That gave us space to place our own things.
Persephone is just a short (but steep) walk down from the upper car park in front of the tiny supermarket "OK" and the bakery "Tsourtis" that is open from early morning to late in the evening.
The village Ioulis is very charming and a good starting point for walks/hikes. You don't need a car on Kea. Just take a taxi if you want to cut the walks into shorter legs. The walks are well maintained and signposted by numbers.
The best map for walks is placed on a large panel at the lower car park of Ioulis. We took pictures of the map and referred to it while using Google maps. It would be very useful to get this map in paper version. But we couldn't figure out where to buy it.
We highly recommend Persephone.