Badlands NP is fascinating, some of the shorter walks would be suitable for your kids. Some of them have boardwalks, I spent about 4-5 hours on a visit and it was enough, I could easily have stayed longer but that amount of time would be ample for younger kids, download the park brochure from the NP website, that will give you an idea of what is there, they have some great pictures also, depending on timing of your vsit you may be able to take advantage of some of the park programs for kids. If you are there in warm weather make sure you take water bottles. It is very dry. Wall is the closest place to the park, it has some accomodation, and a few places to eat,with little kids I would choose one of the fast food places, after having eaten in a number of places in Wall, all of the fast food places would be better.

If you are just in the area for one night and wish to see the park at dusk/dawn, you really need to either stay in the park lodging or directly outside in Interior. Interior is less than a mile from the park so you are practically inside it if you stay there.

The missile site is right off I-90 on the highway leading to the eastern entrance of Badlands. So it would make the most sense to drive from the Black Hills to the park entering either at the Wall entrance or the Sage Creek entrance, then drive through the park to your lodging. Stay overnight, then exit via the east entrance and go to the missile site.

I don't have kids so I can't comment on kid-friendly activities. It looked like there was a pool or playground at one of the places in Interior. They can always wander around in the park but in summer, you really need to watch for snakes even at night. There is a little store on the way to the missile site that has a prairie dog town right in their parking lot, you can buy food and feed them.