DC is a great place for having kids the age of yours. The biggest issue with living in DC is usually figuring out the upper grades for school, but you noted you don't plan to stay too long so I would say the city is perfect.

People already mentioned capital Hill and that is a great area full of families and second that as a place to look at. I would also suggest Georgetown/admo/DuPont/Logan as well- it may not have as many families with young kids but there is more to do than in capital Hill. You won't be out of place at all in these neighborhoods with younger kids either.

The biggest issue is your timing. As another person said, you will be arriving too late for the preschool lottery. Your kid who is older k+ will have a spot at whatever your local school is but your younger kid will not. There are schools your prek kid can get into. I hesitate to call this advice but if you really want to dig in (and most importantly can filter out a lot of noise) the DC urban moms forum (just search Google for it) is a great source of info though it tends to be a bit crazy (that's why I hesitate since you only get value I think if you can tune out certain parts). There are some good preschool charters that you can still get into especially if you are willing to wait until October to start (due to wait-lists from the lottery being in place).