If you want to stay in a great, lively (but still quiet) area with great restaurants and bars which is also very close to all of the tourist sights of Copenhagen then Jakob's apartment is your best bet! It's definitely a special treat to stay in this apartment with its beautiful fifth floor view of the canal and the church of our saviour. It has a lovely modern appearance with many interesting paintings. Beds are comfy even the one in the loft!
Minor con: no lift in building for big luggage.