You will get an absolutely tremendous scran for breakfast at a place called The Union Kitchen - it's in a side street off Nyhavn harbour. Bizarrely, it ended up being listed on my Monzo as Glasgow University Union, despite being unable to find a single connection between the two. Very good, efficient service and friendly staff too. Enjoyed it so much that we went back there another day for an evening meal and got some meatballs, etc - you can choose a few different smaller dishes. It was also an excuse to see Nyhavn all lit up at night, which is a different charm to during the daytime. As others have stated, it's an easy city to walk around and see essentially all the main sights it has to offer in just a couple of days.
I'd say a wee jaunt over the bridge is a must, even if just to say you've been over the longest bridge in Europe and one of the most unique in the world, and also to be in two countries in a day. Plus if you're a fan of the TV series, even more reason.