Lived in Copenhagen for a few years and drove to Berlin a few times including Christmas. Berlin is just a big city and other than Brandenburg Gate, The Berlin Zoo (young kids), it's just another big city with the usual attractions that you can find in any other big city. Remember Berlin was leveled and rebuilt so not too much "old" to see but if you are visiting people there, they will keep you busy. You can still tell if you are visiting the old East of West side of Berlin...
Note - I hate shopping and every Christmas market seems the same to me and I hate them other mulled wine and food...
Visit Potsdam which is 30-45 minutes to the SW of Berlin. Lots of palaces/summer homes in the valley that survived the war and is a nice walkable town- "feels" more "old Germany".
(I second visiting Prague if you can swing it - one of my favorite cities in N Europe...)
The Hartz mountains between Berlin and Hamburg have some neat little towns and attractions including skiing.
Hamburg has a few things to see as well but again, mostly the same stuff as any other German city. (Model train museum is pretty impressive for kids and adults)
Lubeck is a neat small town with a Christmas market and filled with Danes making the run across the border for cheap shopping.
Aachen is cool but totally the opposite direction you are going.
Shop, eat and drink well while in Germany because you will spend out the nose to do the same when you go to Copenhagen. Tivoli is a must stop as is the harbor area and Christania while there. Christmas markets in CPH aren't anything special... (And yes, Noma is worth the experience if you can get a table - but it's at least $400/person w/o booze ...)