You have a range of gym chains in Copenhagen. These include SATS, FitnessWorld and FitnessX. All of them offer student discounts and the possibility to workout at different centres. The monthly prices alternate around 200 to 300 DKK.
We are mostly working out at SATS since we like a good sauna after our workout. But you will certainly be happy at any of the bigger gym chains.
You can also workout at CBS with CBS Mercury. They also offer Crossfit. Check their Instagram for further information: @cbsmercury.