I've got several things to add. I know DC pretty well and we took our family there this past summer.
As to where to stay, it will usually be cheaper, by quite a bit, to stay just outside of DC, rather than in the District, and ride the Metro in each day. If you've not done it before, the Metro is clean, easy, and safe. You're probably safer on the Metro than when you come up top! For example, Georgetown is a great area, but rates will be much higher here and the nearest Metro stop is a bit of walk from Georgetown proper. If you plan ahead a little bit, you can buy preloaded Metro cards for everyone and have them mailed to you before you go.
As to airports, Reagan is much simpler to fly into. Dulles and BWI are quite a ways out. The Metro also goes to Reagan. Since you said you would be using Hilton points, here is what we did this last vacation. We stayed at the Hilton Garden Inn -Reagan National Airport. There is also a Hampton Inn just next door. Both had very good rates, being on the Alexandria side of the Potomac, They have a free shuttle to and from the airport (about a 10 minute drive), or you can take the Metro as it is just one stop away from Reagan. These hotels are in the Crystal City area of Alexandria. There are plenty of eating places you can walk to, the area feels safe for walking around, and Crystal City Metro stop is a 5 minute walk at most. Taking the Yellow line from there, it's about a 10 minute Metro ride to the Air & Space Museum and the Mall. You are just on the other side of the Potomac from The Mall and Smithsonian.
Air & Space would be high on the list for kids. Many kids like the Natural History Museum. One I would suggest that our family really enjoyed is the recently opened new International Spy Museum. It was accessible by Metro also. If you want to do a tour of the Capitol (and we enjoyed it), you will need to go online and secure a spot. If the kids are into zoos, the national zoo is on the Metro, too. We did a nighttime monument tour that we all really enjoyed. There are several different companies that do it. We did the one that leaves each evening from the Navy Memorial on Pennsylvania Ave and was very satisfied. I would also say the Arlington National Cemetery tour was well worth it and all enjoyed it.
If you get a hankering for some Texas BBQ while you're there, believe it not, there is a really good place several blocks north of the Mall, Hill Country BBQ. It was well worth it and was the real thing. Several easy places to walk to for food are near the hotels I mentioned. Big food court at Union Station and in the Metro area underneath the Spy Museum are easy places to take kids (not great food but easy) are other places we grabbed a bite where we could have some variety for different tastes.
I hope some of the above gives you some good ideas to follow up on and decide what would work best for your family. But for ours, staying where we did near Reagan and riding the Metro everywhere was a snap and we would do it again.
ZTGA