It is over 1400 miles from Birmingham (Alabama) to Mount Rushmore near Keystone (South Dakota). It will take more than 21 hours to drive (not including traffic congestion or any food/fuel/rest stops. It's not clear what you mean by 'place to stay'. I'd advise you to use the long-established and reliable https://www.booking.com/ to find hotels/guesthouses/lodges etc etc in and around Keystone and Badlands National Park. The site has excellent mapping, lists almost all accommodation and reviews are trustworthy because they can only be posted by people who have booked through the site and completed their stay.