Try to stay in www.bedincph.dk. It is only 7 minuttes away from the central station. There is a nice kitchen with free coffee and tea. The beds are very good and the location is peaceful.
There is everything you need just around the corner. The hosts are friendly and will gladly help you make your way around copenhagen. You share bath and toilet with one other room...that is the only bad thing we can say about it....but everything was nice clean.
- Room Service