Unfortunately, you cannot deduct the housing your daughter had to pay. It would qualify as a Job Related expense, but this is not deductible for tax years 2018-2025; and it is an miscellaneous itemized deduction, so she would unlikely be able to benefit from the deduction regardless.
She will need to file the Minnesota nonresident state tax first, then she will get a credit on her Illinois state tax return so that she is not double taxed. See Multiple States—Figuring What's Owed When You Live and Work in More Than One State.
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