"We loved staying at Hansen Inn and cabins in Wall, SD. It's such a quaint town and fun for us to be able to walk to Wall Drug and also drive to the Badlands one evening at sunset and again during the day to experience both views.
1hr-ish drive to Mt. Rushmore, it worked out to be a great place for us to stay for 2 nights - two visits to Badlands, a visit to Mt. Rushmore, and when checked-out spent some time shopping at Wall Drug and the other tourist shops. There's also a grassland visitor center right there that we didn't end up visiting. The rooms only have cleaning service if you request, so that isn't a huge deal, but something to be aware of (I think most stay 1 night so it doesn't affect the typical stay). It's nothing fancy in the standard rooms, but the remodel decor makes it quaint. There's an outdoor patio with a cornhole set and fan - I used that area to exercise (we happened to have amazing cooler weather during a summer stay). There's a processing fee for credit cards, fyi."