I went on the BEST free walking tour when I was there!
http://www.tripadvisor.com/Attractions-g189541-Activities-Copenhagen_Zealand.html#TtD
We did Sandeman's, but it looks like there's another highly rated free one too (we did tip the guide at the end).
We were there for 36 hours. Upon arrival, we got on one of the boat tours in Nyhavn, and then went to the Ice Bar (which was a waste, but kind of fun). Then we walked over to Tivoli, and contemplated walking through the gardens and having dinner there, but the entrance fees were high enough that we chose not to. Just walked the main strip of shops instead.
The next day, we went on the walking tour, had a picnic lunch in the park, and wandered around some more. It is an extremely compact city that we could walk back and forth across multiple times without issue.