She'll file an Ohio non-resident return and claim the income she earned in Ohio.
Next she will file a Full-Year resident return for Illinois and claim all her income.
Please do the state returns in that order.
Her Illinois return will give her credit for the Ohio tax she pays on the Ohio income.
Illinois has a higher tax rate, and she will only be given credit for the tax Ohio charged, so she will still owe the difference on the Illinois return.
Since she is a dependent, her resident state is your resident state.