My girlfriend and I had a wonderful time at Nicolaj and Ruiting's place. Small but cosy room, with a private bathroom, and just about everything you need (city info, maps, tv, fridge, wireless internet, a safe place to store your bicycle in the basement). The appartment is quite close to the city center. Bus 5A takes you to Tivoli in a couple of minutes (2-3 bus stops only) while the airport is reachable by the same bus within 30 min.
And we really enjoyed the restaurant recommended by Nicolaj :)