Looks like you can do a 3-3.5 hour loop from Indio going up through the park to Joshua Tree or 29 palms. So you If you left at 7AM, you should have 1.5 – 2 hours worth of stops you could make within the park.

Some quick neat things:
Cottonwood Spring
Cholla Cactus Garden
And I know there are some neat rock formations at some of the campgrounds/near the road.

from http://www.nps.gov/jotr/planyourvisit/things2do.htm

If you have four hours or less, begin your tour at a visitor center where park staff will be happy to help you plan your visit. With limited time you may want to confine your sightseeing to the main park roads. Many pullouts with wayside exhibits dot these roads. There are 12 self-guiding nature trails. Consider experiencing at least one of these walks during a short park visit. On clear days the vista from Keys View extends beyond Salton Sea to Mexico and is well worth the additional 20-minute drive.

A park ranger has never done me wrong.