How do you feel about roughing it? I stayed at Mt Rushmore KOA campground in a 2 room, no plumbing cabin. There were showers and toilets very close by. Location, location, location. Only a few minutes to Mt Rushmore itself, 15 to Crazy Horse. Also very close to Custer State Park. It's about 45 minutes to Rapid City. That would be my second choice for a place to stay if you want amenities....
FWIW, you could easily spend 10+ days there and not do something twice. As mentioned, Devils Tower, Badlands, Wall Drug, downtown Rapid City, Lead and Deadwood, Custer State Park, etc, etc. It is a great area.