I reckon it’s pretty hard to go wrong if you pick a hotel in Mitte based on the likes of TripAdvisor or Hotels.com. There’s loads of choice at all budgets, and you won’t be far from the action.
Personally, I can recommend the Titanic Chausee (crap name but nice hotel) or Novotel – booked through BA Holidays which was incredibly cheap. BA seem to have surprisingly good deals.
Air BnB is worth a punt, but we’ve rarely found it great value against even good hotel deals.
As for things to do, there have been posts on here before. Berlin do’s and dont’s But the Budestag / Reichstag is worth a visit.
If you’re old enough to remember life before reunification, the best thing I’ve visited is the old Stasi remand prison in the East of the city at Hoheschoenhausen. It is incredible. Moving, thought provoking, fascinating and bizarre. Just make sure you line up an English tour (unless your German is up to it, in which case you might get one of the old inmates to show you round).
Stasi Prison Museum