It sounds like the state is in Box 15 is Wisconsin for the $4,671 withheld in Box 17. If that is the case, you should file W-220 with your employer so they stop withholding Wisconsin tax, and instead only withhold Illinois tax.
The withheld money from Wisconsin can't be applied to Illinois. In order to get a refund of Wisconsin tax, file a non-resident tax return, reporting zero income as Wisconsin sourced, and request a refund. For Illinois, file a resident return and report the wages as if you earned them in Illinois.
Here is more information on reciprocal states: What is a state reciprocal agreement?
