I highly recommend staying at Ulrik's and Sara's place. It's a huge and very cozy appartment. Everything was superclean and there's a beautiful, quiet balcony. Although we didn't meet Ulrik and Sara in person, they took very good care of us, feeding us with all the information we needed. and some beers (thanks a lot!). Also Ulrik always replied very quick on my texts and emails.
The neighborhood is very nice, delicious bakerys and great coffee shops around the corner.
If I ever need a big appartment in Copenhagen again, I'd love to come back there.