HFS backlog in Cook County (Daley Center) is about 6-9 months from what I've seen. You can file a motion yourself as it relates to BACK OWED amounts ONLY, or you can get a private attorney to do this. This process takes a week or two, instead of months, to get in front of a judge.
I agree with attorney Gotzh. Hire a private attorney and your case will move much faster. If the father is found in contempt of court, he might be required to pay part or all of your attorney's fees. Or you can just wait for the state. They will get the job done, but the process is very slow.