We stayed a couple of nights at Cube Suites. First impressions were great - Yusuf met us at the door, helped with luggage, offered us tea (which, to my shame, I never took him on on), showed us round the apartments, etc. We had two on the same floor - one for the kids and one for us. This arrangement worked perfectly for us: the kids are 8 and 4, and are quite capable of navigating a few feet from one door to another if something goes wrong.
The apartments themselves are comfortable and nicely decorated. The one on the front of the building did get a bit of street noise, but closing the (double-glazed) window and using the a/c meant that we weren't bothered at all with it while sleeping. We did sip a glass or two of wine on the roof terrace after dark as well.
The location's lovely. A bunch of good restaurants, a supermarket and a park with a playground are within a few minutes' walk; the sights of Sultanahmet within 10 or 15.