You only have to pay tax to Michigan. Illinois only taxes income "paid in Illinois," so income earned while working remotely in Michigan is not "paid in Illinois" and not subject to IL income tax.
Even if it were, Michigan and Illinois have a reciprocal agreement, whereby a MI resident working in IL pays tax to their home state (MI) and not their work state (IL). Your employer should be withholding MI tax. If not, file IL-W-5-NR to stop IL withholding and MI-W4 to start Michigan withholding.
See When is compensation paid in Illinois? in Who is Required to Withhold Illinois Income Tax.
