More sugestions:

Check the web for guided tours. That may give you a diverse impression of COpenhagen.

If you want to be on your own visit
Rosenborg Castle with the Crown Jewels.

In Roskilde, 1/2 hour by Train, you have Roskilde Cathedral where the kings & queens are burried.

In Roskilde you can also see Viking Ships.

Tivoli will be closed in January.

Note that the VAT in Denmark is High (25%) However, look for "Tax-Free" shopping sign in stores, and you can get 18-20% back when you leave Denmark for a country outside the EU. Have the receipt stamped as exported by customs at the CPH airport before going into the transit hall. In the transithall they have an office in which you can get most of the VAT you paid back.

When I was in DK a month ago, I visited Royal COpenhagen's store in the Transit Hall.
Placed my order for pick up on my return to the USA. All ites were giftwrapped and ready as X-max gifts Remember that purchases in the Transithall is w/o VAT.