Most of the above suggestions are way above your budget. Here are a couple of comfortable hotels in my neighbourhood, which are within your budget ballpark and also meet your location criteria:

Hotel Jurine
and the Kastanienhof

Of the two, I would say the Jurine is nicer. It's a family-run hotel owned by a French fellow and his German wife. Both places are in Prenzlauer Berg just on the border of Mitte, sort of eastern Berlin's Greenwich Village, with oodles of hip restaurants, cafés and bars in the immediate vicinity, a sort of budding fashion/artist's district on Kastanienallee and Zionskirchplatz, a fast tram line, a nearby U-Bahn (subway) line, the trendy Hackesche Markt district with its stylish bars, restaurants, boutiques and clubs only 15 minutes walk or 5 minutes tram away, the equally stylish Kollwitzplatz neighbourhood also about 15 - 20 minutes walk, and fast access to most of the principal tourist sights ie Brandenburg Gate, Museum Island, Reichstag etc and the Friedrichstrasse shopping mile.