If you're on public aid, contact DHS, they will prosecute a child support proceeding on your behalf for free. Only the father's income is factored under Illinois' child support guidelines (750 ILCS 5/505), you will be unable to compel the wife to contribute anything to a child that's clearly not hers.
The Department of Human Services to make a child support argument and petitions on your behalf.
Family Community Resource Center
600 S State St
Elgin, IL 60123
Phone: (847) 931-2700