We spent 8 days in the Black Hills in June. We were probably ready to go after 6, but you could definitely find lots of things to do there to use up most of your time. If you did want a 'beautiful mountain' portion of your trip, consider a couple days in the Big Horn mountains near Sheridan/Buffalo WY. It's gorgeous there, but not nearly as far as Yellowstone. Could see Devil's Tower on the way over.
Also there's not much to do in Cheyenne. I'd try to limit that to just 1 day so you can spend more time in the more "fun" areas.