Illinois has a procedure for a hardship permit due to a child support suspension but you still have to get something from the family law court to show you are dealing with it. Go to the Danviille DMV for a consultation to find the exact steps. But the SOS cannot use the permit in the face of the suspension.
In addition to the advice given by Mr. Harvatin, contact the DMV in Florida and see what may be needed to clear any suspension/revocation/hold/etc. that may be on your FL license (presuming, of course, that you resided and held a license in FL before you left the country for 7 years). If you did have a FL license in the past and cannot apply for your IL license b/c of a FL sanction, contact a FL attorney and see what can be done and/or what documents may be needed in FL to clear up that action. Good Luck!!
Some things are simple, but not easy. That's what you have here. You say the child and the custodial parent are here. Bingo, Get the "D" or "F" case, whichever it is, transferred to your venue. (Any Domestic Relations/Family Lawyer knows how to do that--it's pretty simple). Then you follow Ted's advice and you have an Illinois Court Certifying you have an installment agreement. You comply with the installment agreement. You might need to share the whole mess with FLA but they will drop you like a bad habit when they see they are no longer protecting a FL child.
You have to do it orderly, though. It took you a long time to make this mess. It'll take a little while to get it cleaned up. I'm glad you're excited about a specific job selling cars but, I promise you, there'll be another one when you get your license cleared.