This is not difficult, people.
...It just isn't well publicized. This comes up over and over again when working with new landlords and with agents who don't do a lot of rentals. Chicago puts a limit on the late fees that can be charged to tenants who don't pay rent when its due. (Oh, and for the record there is no state mandated grace period in Illinois. If rent's due on the first, it's late on the 2nd unless your lease says otherwise. End of story.)
So if your apartment is covered by the CRLTO the maximum you can charge is:
No rental agreement may provide that the landlord or tenant ... agrees that a tenant shall pay a charge, fee or penalty in excess of $10.00 per month for the first $500.00 in monthly rent plus five percent per month for any amount in excess of $500.00 in monthly rent for the late payment of rent;