No, your daughter is not a Minnesota resident. In general, you do not become a resident of a new state when you are there for a temporary time. (and she also needs the intention of not going back to her residential state).
She will file a resident Illinois and a nonresident Minnesota. She will be able to claim the credit for taxes paid to another jurisdiction on her RESIDENT return (which is why the part year resident designation you selected makes a difference.)
See Who is an Illinois resident? (This may be helpful for when she goes to college if she goes out of state!)
If you have any other questions, please ask!
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