We can highly recommend Elenas Appartment! It's perfectly located next to the main square of Kos. The streets around are full with nice boutiques, cafes, bars. The appartment itself is spacious and provides everything you need for a stay in Kos. Elena made us feel welcome with cakes, wine and water. Very nice! Communication was very good, quick and helpful.
Kos-Town is worth a visit. There are some very touristy parts in the old and near the harbour and also a "Party-Area" which caused some noise at night. But overall its great to walk around and enjoy the atmosphere. We also took a bike for a trip along the coastline.
So we recommend Kos-Town and Elenas Appartement!