Two suggestions: both can be done in 8 hours. Copenhagen city is very compact, so getting around never takes very long.
Boat tours from Nyhavn - take the tour boat at the top of the canal (you'll know what I mean when you see it). It runs like a taxi service, you can get off at a whole load of different places including the Little Mermaid, Trekroner Fort, Opera House, Christianshavn (which I think is a nice one - you can walk to and climb atop Vor Frelsers Kirke - google it! incredible view of the city) and plenty more stops besides. Or you can stay on the boat for the duration and listen to the audio info on the various sights.
The second thing I would recommend is Tivoli. The magical amusement park so wonderful Michael Jackson repeatedly tried to buy it. Not just for the kids. Full of magic plus some a variety of restaurants, at least one Michelin star joint, it boasts its own brewery and plenty of cafes, live music, theatre, oh and a roller coaster or two! Some for the more brave among us, but some lovely quaint rides too for the jitterier types. about 6 years ago