Welcome to RTA!
The Denver area is surrounded by gorgeous mountains. There's so much to do there! North of the city is Rocky Mountain National Park and the beautiful Trail Ridge Road. South of the city, in Colorado Springs is Pikes Peak, Garden of the Gods, and the Air Force Academy (the Chapel is beautiful). East of the city on I-70 is Vail and the routes to Aspen, plus Glenwood Springs and in Glenwood Canyon, the "hanging freeway". You'll have no problem filling a week! Allow at least a day to get from KC to Denver -- depending on where you start in the KC area, you're looking at 605 miles.
It's about 375 miles from Denver up to the Rapid City area where Mount Rushmore is located. From there, it might be "easiest" to just stay on I-90 and cross South Dakota. You could stop at the Badlands, Wall Drug, the Mitchell Corn Palace, and the Chamberlain Rest Area (which is full of Lewis & Clark information and a beautiful view of the MO River), if you have the time. Bear in mind that it's 2 days from Mount Rushmore back to KC, MO, via I-90 to I-29.
A day to drive to Denver, 7 days around Denver, a day to drive to Mt Rushmore, and 2 days to drive home takes you to Day 11. So it doesn't leave a lot of extra time on that return trip.
Donna