I've visited Berlin a few times in the last 10-15 years; although never with an infant, so take with a grain of salt.
First, staying near Gendarmentmarkt makes for an easy walk across the river into the Museum area/island. If you can only go to one museum, go to the Pergamon. Very visually stimulating for the little one so they will have lots to look at and you will be amazed. Truly a gem.
I've stayed at the Hilton on the Gendermanmarkt Square during the holidays and they have one of the largest and best Christmas Markets in the city. Not sure how late in November, but check the holiday market schedules. That was actually the entire purpose of our last trip: Christkindlemarkts all across the city and in Dresden.
Right off the market square is a giant chocolate shop. Go upstairs and get the most expensive and best hot chocolate you will ever have. I forget the name of the place, but you can't miss it as it is across the street from the Hilton and underground metro.
Berlin also has a great zoo, although weather and outside light could be an issue for you. It gets dark early that time of year, and the weather/cold could be an issue.
I love, love, love Berlin. I am sure you will have a great time enjoying a very eclectic and vibrant city. Have fun.