Just a suggestion -- you might think about staying in Custer -- neat little town and the drive from there to the Mt Rushmore Monument is one that is fun and Beautiful with many different route -- Custer State Park is not to be missed along with the Caves -- Jewel and Wind -- The Black Hills deserve more than just the one day stop so plan your visit to include many of the sites that are available.
We stayed at
Custer's Gultch great spot -- while many think you need a CG with lots to do -- IMHO -- there is so much to do in the area you will only sleep at the CG.
JMHO,