I think you can drive straight through Chicago to New Mexico in 25 hours or less including quick rest stop breaks.

Main route appears to be I-44, I-40, US-60, US-70

If you have time for a stop in Oklahoma City (zoo?), the adults will want to take a look at the bombing memorial too.

Carlsbad Caverns - kid friendly? Hmmm... There were children on the two tours I took - but not as young as 2 yrs. Some got a bit scared of the closed & darker spaces. But other kids thought it was all pretty neat! I'd call the tour reservations line to ask if any tours are kid friendly. There often are narrow trails, dimly lit, with nobody allowed to touch the rock on either side (for preservation purposes). Some trails also have a lot of stairs or ladders to climb. Also, you need to walk in a straight single file line down the paths & keep up to the guide. This can make it pretty hard on a toddler. I don't know if any of the trails would fit or allow strollers. Call to check.