Yes, you will file in both states because going to school is considered a “temporary absence” from your state of residence so your daughter remains an Illinois resident.
File a nonresident return for Missouri and a resident return for Illinois.
Owning $322 to Illinois appears correct. IL has a flat tax rate of 4.95%, which comes to $322.
States normally give you a credit for tax paid to another state, but none of your daughter’s income was taxed by MO so there is no tax credit on the IL return. You would get a refund on any MO withholding.
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