My main recommendation for Berlin is to book the longer guided tour of the reichstag. Its free. You need to take your passport. Its fabulous. You need to book it well in advance. Try now. If you cant get in then book the shorter tour. Its fantastic. You and your dd will love the view inside the dome.
Must see the east side gallery. There's a food market called the
Markthalle Neun near there which you
might enjoy if you like that kind of thing.
I think the DDR museum is fascinating. Obviously there's lots of history things your DD might want to see. I found the information about life in East Germany and under communism really different. Book ahead.
Day trip to potsdam was lovely and easy by train. We got the tram from the station to the centre of the town, hit the tea room for coffee and cake, as it was raining. Poked around some lovely shops which woukd appeal to you and dd then walked to and around the unesco heritage sanssousci pparkland then hiked back to a closer station for the train home.
Like you we went to the zoo. Its a great day out. The area around the zoo is amazing for shopping. There's a shopping centre called I think bikini? That teens are meant to love. My 15yo dragged me around many shops in the area (like an even bigger Oxford Street) and we went to hard rock cafe for dinner. I wanted to see the U2 trabant. The kids love hard rock.
There's also a free lunchtime concert at the berlin Philharmonie which I decided to do instead of seeing the berlin symphony orchestra. Because it's shorter and my dc would have no patience for it.
We loved the Rausch chocolate shop in Mitte. There's a little cafe downstairs and a bigger cafe upstairs plus models of berlin attractions in chocolate. Must call by if you are staying near there just to see the models. They sell expensive but very delicious chocolate and the hot chocolate and the cakes and biscuits are to die for.
Next door to the rausch chocolate shop in mitte is the hilton and I think you could pay to use the pool and sauna if you fancied a treat. It's only a small pool but quiet and lovely and there is a sauna and steam room. It says textile free in the sauna and steam room but I went in in my swimming costume!
Do you fancy a trip up the TV tower? The height scared my younger ds and we'd got our ariel view of the city from the reichstag.