I'd say it's less than 5 hours from Cheyenne to Mt Rushmore. Are you actually staying near Mt Rushmore or in Rapid City? I drove from Denver to Rapid City in right around 6 hours this fall.
I second the vote for
Bear Country USA. I thought it might be kind of lame, but it was really pretty cool. Personally, I wasn't all that impressed by Crazy Horse. Kind of disappointing, plus you have to pay just to get to the visitor's center and then pay again if you want to get up to the base. Mt Rushmore was VERY cool, IMO. There's a nice and fairly low-key (although lots of steps) hike that takes you right up along the base. You can rent an audioguide that talks you through the "points of interest" along the way. I loved Mt Rushmore! On our way home, we stopped in Hot Springs, SD at the
Mammoth Site, which was really interesting. There's a free park overlooking Rapid City called Dinosaur Park that was worth a quick trip to. It has huge statues of different dinosaurs and was built during the Works Progress Administration. The view from the park was pretty cool, too. And if you have time, a drive to the Badlands at either sunrise or sunset is definitely worth it! We also did a chuckwagon dinner show at a place that has a lot of the set buildings from Dances with Wolves, which was kind of fun.
Hope that helps a little!