Berlin has great public transport between the s-bahn and the u-bahn. Don't feel as if you have to stay super close to where your wife's friend may be. Most of the government offices are in the Mitte district. I am not sure many people live in that area--unless uber rich. That district has a lot going on, but that is not the part of Berlin that interests me most. But for sure, you want to go to Mitte and sight-see. A nice walk would be from the Brandenburg Gate to Alexanderplatz. The Jewish Memorial is very near the B Gate and is a must. Museum Island near Alexander Platz is great for perhaps a full day. You can get passes. If you have time to kill, I might even take the bahn (I can't remember if it is an S or U stop) and get off at the Victory Column. And then walk all the way to Alexander Platz. It is probably about 3 miles but you will pass some cool stuff.
If I remember correctly, part of the agreement on the wall coming down was that the west would not remove the war memorials and statues of the east. Many of these art installations are on the walk from the Victory Column to the Gate. From the Gate to Alexanderplatz is lots of shopping and such. There is are a couple of Berkinstock stores along this path if your family is wanting to score some of that gear. I always end up bringing back several pair although I don't have any myself. In fact, the Birkenstock store that is on Friedrichstrause (cross street of the Unter den Linden--the road you will be walking down) is very close to a good spot to stop for lunch. The Treffpunkt Berlin is a smallish pub/restaurant on Mittelstrause--not sure how to type that odd German B that makes the "strauss" sound." Nothing super special just a classic German lunch menu and some good beers. I always get a kick out of the signs labelling the genders of the restrooms downstairs. I'll leave that Easter egg for you to find.
I'll log back in later and hit you up with a few more of my recs.