Originally Posted by pbarnette

Personally, I would skip the Mermaid. Not interesting enough if you ask me. I would take the earlier flight and do the following:

1) Train to Central Station.
2) If open, wander around Tivoli for a bit. Maybe grab lunch in Tivoli.
3) Walk through Radhuspladsen (City Hall Square) to Strøget, the pedestrian street.
3a) If you are into design, there is a design museum in the area, or if into sculpture, there is a sculpture museum near Tivoli.
4) Walk down Strøget - the first section has little of interest, but when you hit Amagertorv, I would pop into Illums Bolighus and look around. A shop, yes, but a very pleasant one with a nice array of Danish and Nordic design. Housewares on the first couple of floors and furniture and lighting further up.
5) Window shop the rest of the way down Strøget. Grab a hot dog from a vendor if hungry - I like franskedogs, but rød pølse is more traditional.
6) Walk over to the palace area and mill about. Maybe check out the Marble Church. Walk over to the waterfront and see the opera house.
7) Take a canal tour if that strikes your fancy.
8) Grab a beer (Tuborg is the big local beer - you will likely also see Jacobsen, which is Carlsberg's craft beer line which is very good) at one of the outdoor places at Nyhavn.

For dinner... Assuming you don't want to or can't do noma or Geranium (both will be 3-4 hour meals, very expensive, and excellent), I would consider one of the following...

Geist - right on Kongens Nytorv. Modern Scandinavian cooking. The chef had a star at his old place but shut it down to open the more casual Geist. Inventive and one of my favorites.

Koefoed - I love this place. Very good service. Everything is from Bornholm. Modern with enough creativity to keep you happy, but not as inventive as Geist. Near Kongens Nytorv.

Kødbyen Fiskebar - You will want to get in a cab from Kongens Nytorv, but this is a great seafood place. Very regional in produce and preparation. One of Rene Redzepi's (noma) favorite places.

Relae or Radio - Have not been unfortunately. New Nordic. Rave reviews for both.