Here is the description of my day in Copenhagen in 2010, while on the DCL Northern Capitals cruise. It's in my cruise report which is linked in my signature below:
"We met for our shore excursion at 9:15. It was the “Highlights and Open Air Tour of Copenhagen” excursion. This was essentially the same tour as in Oslo. We rode in a bus and saw several historical sites and got out at a couple of places. We walked through the Parliament grounds (but not inside any of the buildings), and saw the Queen’s state offices, where she has official meetings with foreign dignitaries. We also walked through the gardens of the national library, which is adjacent to the Parliament building. The gardens used to be completely underwater and there is still a large metal ring where former royalty used to tie up their boats when they visited. We also saw the modern wing of the library, which has been called the Black Diamond building. (Its design was pretty controversial, when first presented; but it’s not horrible.) At another stop, we walked through the palace complex of Amelionborg. It’s an official residence of the Queen and Crown Prince. There is also the Marble Church in the complex. Unfortunately, we didn’t have time to walk through the church. The final stop of the tour was an open air museum like the one we visited in Oslo.
[The Open Air Museum is basically a village where old buildings from around the country have been brought and restored. Kind of like Old Sturbridge Village or Colonial Williamsburg in USA] It was mildly interesting, but I didn’t love it, personally. This is where we had lunch, as well. (The pickled herring was not good!) We got back to the Magic at about 4 pm."
I highly recommend visiting Amelionborg and the Marble Church. That was very interesting to me. I believe the "Little Mermaid" sculpture is located in Copenhagen, although when we were there, it was on loan to a museum in China and we only saw it on a big TV screen. Some other things that we didn't get to do, but which were discussed by Disney's guest lecturer:
Carlsburg Brewery -- Apparently, some great architecture. Probably get some samples, too, if you're a beer connoisseur.
Rosenborg Castle -- For amazing gardens
Museum of Danish Resistance @ The Citadel
Going along with that final option, I highly recommend that you do some quick Google research into the Danish effort to smuggle Jews out of the country during WWII. Here's a good place to start:
http://auschwitz.dk/denmark.htm