Honestly, your post just breaks my heart. I can't imagine being in a situation where your child is ill and on top of that worry and stress you've got medical bills and no insurance.
I really don't have anything to offer since I suspect all state laws are different. However, I did find this about Virginia FAMIS, which evidently is low-cost children's medical insurance. Like Medicaid there are income requirements that may not be applicable to you, but if you're not already familiar with it it might be worth a call. And if you're not eligible, maybe they could guide you in some other directions.
http://www.famis.org/welcome.cfm?lang=English
I'm so sorry you're going through this.
-nab