First off (take this in the spirit of helpful advice, not being bitchy) you would probably get more response if you posted in the travel section, and people have given advice to similar questions in the past, so reading through those would probably be helpful.
Now to your questions.
It sort of depends on what your kids are interested in, but from Brandenburg Gate it is a VERY short walk to the Jewish Holocaust Memorial, then on to Potsdamer Platz (most interesting IMO for the photos there near the sections of wall, which show how the place looked while Berlin was divided), then from there to the section of wall at Topographie des Terrors. Read up on that area before you go, because it is far more interesting if you know the history of what you are looking at, where an escape across the wall was made, etc. (though there are signs in German and English explaining things).
All of these things are very close together.
Back behind the Holocaust Memorial is the location of Hitler's bunker, if that is of interest. To discourage Neonazis and Hitler worshippers it is deliberately NOT a memorial, is unmarked, and is used as a carpark, but it is there nonetheless.
There is of course Checkpoint Charlie on Friedrichstrasse, but do be aware that the 'checkpoint' you see there is actually a replica of the original. The original is actually at the Allied Museum (Alliertes Museum) at Clayallee. It also has chunks of wall, an East German watchtower, one of the planes used in the Berlin Airlift, etc. In what used to be the American Army cinema there is an interesting display about one of the spy tunnels built during the time Berlin was divided, though I'm not sure if that is open - it was undergoing repair last time I was anywhere near.
To get there, take the U3 underground to station Oskar-Helene Heim, then from there it's a short walk. On the way you will go past, on your right, what had been the headquarters of the Luftwaffe in WW2, which was then taken over by the Allied command after WW2, and has now been converted into apartments. Though the massive American consulate next to it still belongs to America...
Indoor things in case of bad weather
- Bunker tour at Gesundbrunnen (think this has to be booked online in advance)
- Technical Museum, this includes the building next door, called Spektrum, which is filled with kid-friendly interactive displays
- Former Stasi headquarters
Random
There is a spy museum, and a Stasi museum, neither of which I have visited, so I can't vouch for them
There is a former Stasi prison that can be visited at Hohenschönhausen and a former Nazi concentration camp at Sachsenhausen. Both can be reached pretty easily. I have visited neither, as I don't want to do that to myself, but they are there...
Food
The legacy of former East and West Germany means that certain types of cuisine are available that you might not be able to get so readily elsewhere.
East Berlin had lot of Vietnamese guest workers who were left jobless when the Berlin Wall fell. In the following scramble to find jobs, a lot of Vietnamese opened restaurants. Look for a place offering ONLY Vietnamese food, as there also places offering 'Thai AND Vietnamese AND Sushi' who are obviously just trying to stay afloat. Kantstrasse near your hotel has quite a lot of Vietnamese restaurants along it.
There are quite a few places offering Georgian (county obviously, not the region of the US) cuisine. Food tends to have a lot of vegetables, and usually not too spicy.
West Berlin brought in a LOT of Turkish guest workers, who settled down in Berlin even though they were not meant to be permanent. Of course there are a lot of döner places, but there are also a lot of proper restaurants. Hasir is one such chain, again with a lot of vegetable options.