CPH is a great city for short trips. We go there frequently for the weekend or for special events.

In the city, things I would recommend seeing in order of preference for must do on a short trip to nice to do if you have time:

1. Canal boat tour around Copenhagen. Lots of companies offer them from Nyhavn. Take your pick. You might want a covered boat depending on the weather. You will see the Little Mermaid on most of the canal boat tours.
2. Tivoli. Lovely gardens.
3. Stroll down the walking street (Stroget) to people watch and shop.
4. Amalienborg and the Marble Church behind it. There is a changing of the guard but it is not like Buckingham Palace in London. Still could be interesting though.
5. Walk up the Round Tour (Rundtarn).
6. Take a walking tour of Christiania (Put your camera away for this one)
7. Drive or take the train to the Louisiana Museum of Art just north of Copenhagen.
8. See Kronborg Castle (aka Hamlet's castle) in Helsingor.

That's my standard bucket list when we have people coming from the US to visit who want to see Copenhagen.

As far as places to eat, I like LeLe in Vesterbro. I have also eaten at Herman, one of the Michelin starred restaurants. It was excellent. There is also a bakery and hot dog stand attached to the Nimb Hotel and Herman restaurant called Andersen Bakery. They make GREAT hot dogs (Danish style) and excellent chocolate milk (which is what you drink with your hot dog). You can get to it from inside Tivoli or from outside Tivoli - it is directly across from the train station.

As far as whether or not the Copenhagen card is a good value...I can see that it costs 249 kroner per person. Attractions 1, 2, 4, and 5 above are covered by it. Entrance fees to these four attractions alone will cost 250 danish kroner. And the Copenhagen card also covers metro, bus, etc. So it definitely seems like a good value if you know you will do those things.

Enjoy Copenhagen!