We took our two boys to the Badlands and Black Hills last summer and had a wonderful time. Our kids are older though, 10 and 13, so it is probably a different experience than you might have. I agree with the poster who said avoid the weeks before, during and after Sturgis. We went the week before thinking it would be o.k., but a lot of the bikers arrive early. It wasn't a problem for us, but there was a lot of increased traffic and there were bikers EVERYWHERE. My boys turned it into a game--counting helmeted riders vs. non-helmeted. You would probably have trouble finding camping sites during those times. I've heard it gets really, really hot in the Badlands in August, but we actually had record-breaking cold temps when we were there. The last week in August they lower prices, but some of the attractions close early.
We also live in the Twin Cities. On the way there, we stopped in Sioux Falls to have lunch with a friend and there is an amazing park on the Falls that was a fun break and opportunity to stretch. We ate lunch in downtown Sioux Falls in a diner that's a converted old train car. That is fun for the kids. We spent the night in Mitchell to further break up the drive. My boys really enjoyed the Corn Palace and Corny the mascot, but you don't need to allow a lot of time for that stop. It's quick. They work on the palace throughout the summer so it might not have much decoration completed early in the summer. They were still working on the sides of it in early August last year.
On our second day we had lunch at Wall Drug. I think it's overrated and a tourist trap, but the kids enjoyed it. It is really crowded. Be prepared for lots of souvenirs and tacky stores. The boys did have a lot of fun crawling around in a relatively newer area in the back (a little park with a jackelope statue, an old covered wagon, etc.) and enjoyed "mining" for jewels (at an extra cost). We spent the afternoon driving through and hiking around the Badlands. There are both easy and difficult hikes. If you want to hike, I would recommend a sturdy backpack for one of you to carry the baby.
In Rapid City, we went to the Dinosaur Park. Another very quick stop, but the boys liked it and there are amazing views since it's set on top of a hill. It is just a bunch of large dinosaur statues. Our boys LOVED the Reptile Garden, which is between Rapid City and Mt. Rushmore. In fact, we went to the Reptile Garden and then Mt. Rushmore, and our youngest said at Mt. Rushmore, "I didn't realize a reptile garden could be so much more interesting than a national monument." Mt. Rushmore is beautiful and an amazing feat, but I didn't find it particularly kid-friendly. Our kids enjoyed driving by it more than actually visiting it. We had lunch in Keystone, the little town right outside Mt. Rushmore. It is a delightful little town and we found a kid-friendly deli (Teddy's?), but beware--it is adjacent to an old-fashioned candy store.
Other things we did included a tour of Jewel Cave and a drive through Custer State Park. I am claustrophobic, and I was fine with Jewel Cave because it is big and fairly well lit. However, it might be a little dark and scary and too long of a tour for younger kids. Custer State Park is full of bison who often stop traffic when they congregate on the road. I get our drives mixed up, but I think we also saw prairie dogs and wild burros. We definitely saw those things fairly up close and personal at some point on the trip!
We also went to the mammoth archeological dig site. It was fascinating and the boys loved it. Your kids might be a little too young for it, but it would depend on your older daughter's interest in archeology and that kind of thing. In the same area is a hot springs water park (Evan's Plunge). Don't expect glamorous though; it's very basic. The water feels wonderful though. There is a small, shallow play area for younger kids, but most of the water is probably too deep for your kids, unless the older one is a really good swimmer. Life jackets are an option though. I saw a lot of little kids there wearing life jackets.
Although we were very busy every day, we were unable to see and do everything in 5 days. There are a lot of fun options. I'm not as familiar with the stuff that appeals to the much younger kids. Many of these attractions do require some drive time, so you need to factor that into your plans. We stayed in Rapid City so we were fairly centrally located, but most things are a 30-90 minute drive away. Feel free to send me a message if you have any specific questions. Have fun! (Book soon--spaces do fill up since it is such a short tourist season for them.)