I would stop at Devils Tower on the way to the area and spend the night there, it is on the way. I prefer National Forest or state park campgrounds so there are many in the area. I prefer to stay at Horse Thief Lake Campground just south of Rushmore.
I would include either Wind Cave or Jewel Cave, Mt Rushmore, Needles Highway and the wildlife loop in Custer State Park, Highway A from Custer State Park to Keystone (this road has some great scenery and some unique pigtail bridges and tunnels that if traveling north-frame Mt Rushmore as you drive through them, great photo op)
After spending your time in the Black Hills, spend 2 days at the Badlands and do some hiking, sight seeing, climbing carefully in the park. See the Minuteman Missile site, and Wall Drug on the way out. I would also stop at the South Dakota Air and Space Museum at Ellsworth Air Force Base just east of Rapid City on the way through.
16 days would be about perfect for this trip.
I do the same type of trip from Wisconsin every couple of years. Next time I go I will be hiking to the top of Black Elk Peak (formerly Harney Peak) and riding my bicycle on the Michelson bike trail. That may add to much for your 16 day time limit.