Did you check the box for non-resident state and fill out non-residency worksheet on federal info worksheet? 

There should be NO tax credit on Illinois. These are reciprocal states. You pay full tax in illinois on the income and claim a full refund on Michigan. 

What I always try to do is file the non-resident state first and early and use the refund to pay off illinois. The bad thing is is that there will be a penalty for underpaying Illinois income tax as no tax was withheld. If employer does not withhold Illinois going forward and client can afford it, make estimated tax payments.