A Slightly Roundabout Option
The most direst route is pretty much I-80 and it would be a bit over 2,100 miles to Chicago. But with 6 days (5 nights) I think you could add just a few hundred miles and get a quite intriguing drive. Start out on I-80, taking a short time out to see Lake Tahoe be3fore heading out across northern Nevada which will have long stretches where the scenery doesn't change much. From Wells head north on US-93 into Idaho and then use I-84/I-15/US-20 to Yellowstone. Leave Yellowstone on US-212 to connect with I-90 for a stop at the Little Bighorn. Make a short detour before leaving Wyoming (again) on US-14/WY-24 to see Devils Tower. Then continue east to Mt. Rushmore and the Badlands. Some other possible stops between there and Chicago include the Corn Palace in Mitchell, SD and the Circus World Museum in Baraboo, WI.
Go ahead and map that out and see if you don't think it would appeal to your family. If you take 6 days for the journey, you'd 'only' have to cover 400 miles a day which would leave at least some time for getting out of the car and taking short hikes around the various places you'll be visiting. And yeah, I think I'd try to work out a workable schedule and book ahead so that you're not wasting precious vacation time on trying to find a bed each night.
AZBuck