What to do in Copenhagen in 2 days

Take a free walking tour, you can find them on the internet.

A canal cruise is nice. You find them at Nyhaven.

There are some castles worth visiting as well as the Royal Palace.

Rosenborg Castle - make sure to go into the basement to see the royal jewels.
Papiroen - a very unique dining experience
Christiania - a community of free spirits
Climb the steeple of the church a couple of blocks away from Christiania...you can't miss it
Amielenborg Square - changing of the guard and the royal palacaes
Take a free walking tour from the tourist office near Tivoli Gardens.
If the weather is good take a break and have a cup of coffee at the small pier just next to the opera house on your left as you're facing the water...super modern design with free deck chairs to sit in.
The garden at the back of the Museum of Danish Design is also a beautiful spot.
Look inside the city hall building - won't take long but it's worth seeing.
Take a boat tour of the canals from Nyhavn.
Walk up and down Stroget - the main shopping street.

All great suggestions.
However, Papiroen, which was a big food stall indoor market; has now permanently closed as of Dec. 22.
There are plans to open somewhere else around May this year.
There is a wonderful indoor food market called Torvehallerne.
It's quite expensive, but a joy to walk around and see all the various stalls.

Amielenborg Square - changing of the guard and the royal palacaes

Show up at the barracks in Gothersgade at 11:27 and follow the guard while they march through the streets to Amalienborg, where they change the guards.

The At a Glance link for Copenhagen gives Rick's star-rating to the list of typical (mostly indoor) sightseeing destinations. Focus on the top ratings and those that are easiest to group together, since you can't do it all.