I would highly recommend staying at Birger's apartment, especially if you want to be close to the main city attractions, but far away from the tourist crowed and noise. Here the streets are wide, the air is fresh (so close to the waterfront) and you can feel the local atmosphere and get know Danes a little bit.
It is so nice to have lunch or breakfast in one of many in the neighborhood coffeeshops and cozy restaurants.
Definately the "hygge" experience.