you are a Michigan resident earning salaries, wages and/or commissions in states having a reciprocal agreement with Michigan (Illinois, Indiana, Kentucky, Minnesota, Ohio, and Wisconsin) you are not required to pay tax to these states. Michigan residents working in reciprocal states should claim an exemption from that state's income taxes.
Q. Which state do I owe taxes in?
A. Michigan
The reciprocity rule applies to your situation (a student temporarily in IL for short term job). You are still a MI resident for that period of time. Michigan residents working in Il should claim an exemption from IL withholding from their pay, If IL tax was mistakenly withheld from your pay, you will have to file an IL return to get a refund. MI will not give you a credit for tax paid to a reciprocal state.
For others reading this:
This is the general rule: The income is work state (WS) source income since it was earned there. Resident States (RS) tax all their resident's income, regardless of where earned. You will file a non-resident tax return for the WS and report the WS income. You will file a full year resident return for the RS, reporting all your income. The RS will give you a credit, or partial credit for any tax paid to the WS. The general rule does not apply in a reciprocal state situation.
If you're still employed in Illinois, then you should file Illinois Form IL-W-5-NR with your employer. Here's a link to that form: