You should check out Cofoco. Great food at a moderate cost. cofoco.dk
Vesterbro, i.e. the "ex" red light district, where you will find a number of nice places to eat.
Had dinner at Nimb the other night - one of the best restaurants in the Nordic surely! Would love to stay at the hotel but that's a bit pricey.
Have a burger at Halifax. Serious burgers, also excellent vegetarian ones.
Frederiksborggade 35, Copenhagen 1360 K, Denmark
Tel: 33 32 77 11