Was in CPH a few months ago for a few days ... it's a fabulous place to visit ...

I'm not that into castles etc. so spent most of my time walking the city. My highlights ....

Resistance Museum

Walking along the seafront/canals to the Little Mermaid, (great architecture on both sides especially the new opera house & library)

Eating a herring meal on one of the many restaurants on Nyhavn

A visit to Tivoli garden (especially at night)

A slow walk along Stroget ...

I also too a few hours and took the train accross the the new Oresund Bridge that links Denmark (Just north of Copenhagen) to Sweden (Malmo) across the Øresund Sound. The train takes about 35 minutes - Malmo has a beautiful old quarter downtown that is worth a visit. The view from the train on the crossing is not that great - would be better if you crossed by taxi but that could be a little pricey.