As has been said, Tivoli is a must. Critics might say that it's only a fairground with a few extras but it's like no other fairground you'll ever visit. (It exudes 'style', rather than 'sleaze'):
http://www.tivoli.dk/en
There are loads more suggestions on the city's official website:
http://www.visitcopenhagen.com/copenhagen-tourist
However I'd also recommend hopping on a train to Malmo (Sweden), which is just across the river from Copenhagen. You get there via the Øresund Bridge (which is, rather oddly, partly a tunnel under the river and partly a bridge over it). There are 4 trains per hour, with a journey time of just over half an hour and, as I recall, they're quite cheap. While Malmo isn't packed with tourist attractions, there's enough to keep you occupied for several hours and it's an attractive city, with lots of good places to eat and drink:
http://www.malmotown.com/en/