You may want to consider the area around Gendarmenmarkt. It is a picturesque square with many "full-service" hotels in your price range. If you are ok with chains, there is a very well-located Hilton, the Hotel Luc, or NH Collection Berlin Mitte am Checkpoint Charlie to name a few.
There is a good variety of restaurants/cafes close by (everything from a nice-but-not-too-fancy Augustiner pub with traditional Bavarian/German food to sushi), a huge Rewe grocery store for snacks/drinks, pharmacy and DM drugstore nearby, and you have access to plenty of transit options (Stadtmitte UBahn station will get you to more or less everywhere with minimal connections, short walk to Bus 100 sightseeing route). It's about a 5-10 minute walk to Unter den Linden, 10 min to Checkpoint Charlie and about 15-20 to the Brandenburg Gate or the German History Museum/Museum Island area. Berlin is sprawling, so it is hard to be near all the major sights, but this location is very central without being too overwhelming. There is also a wonderful antiquarian shop selling old prints/maps across the from Augustiner restaurant - I always make a point to stop by and browse, the maps and prints make a nice souvenir too.
Otherwise, agree Hackescher Markt area could also be a good option, though slightly less walkable to everything (but good transport connections).