That would give you 5 days to get from the GC to Chicago - because Chicago to Cleveland is a fairly easy 1 day drive via I-90. By fastest route, GC to Chicago is a 3 day drive, so you have some, but not a lot, of time to play with.

Here's one suggestion - leave the GC via the east entrance to US-89, go north to US-160 to Kayenta, then US-163 into Monument Valley. Take a Jeep tour. Take US-191 north and spend the night in Blanding or Monticello.

Take US-491 to Cortez, then US-160 to Durango. Take US-550 (the Million Dollar Highway) to Montrose, then US-50 to Poncha Springs, spend the night in Salida.

Take US-285/24 through Leadville to I-70, go east to US-40, take that over Berthoud Pass to Granby, then US-34 through Rocky Mountain National Park (Trail Ridge Road) through Estes Park to Loveland, spend the night.

Take US-34 to I-76 to I-80 and spend the night around Omaha, then on to Chicago the next day.