It is probably a month or two late for July Mueller reservations. Cheyenne Mountain State Park might still have some openings, but probably not many or for long. I cannot picture a Garden of the Gods RV park, but there are some on Hwy 24 that are close. There is a KOA just south of the Springs on I25. They have amenities, but it is a typical KOA on an interstate. Just a little north in Monument there are parks along the interstate.
The kids will like the Dinosaur museum in Woodland Park. The train through Royal Gorge and Possibly the bridge.
There are some electric spots at Eleven Mile State Park, but like Mueller that is over the pass to the west.
Garden of the Gods, Manitou, cliff dwellings and the cog railway are always accessible from anywhere in the area.
Edit: The Academy (along with other DoD locations) has changed requirements, so it is likely without a DoD ID card you may have a wait to get on base. Updated info is here: http://www.usafa.org/Connect/Visiting