You can file a Non-Resident Return for New York, since you still considered yourself an Illinois Resident during the months of your internship.
Prepare your Non-Resident return first, then your Illinois Resident return. You report all your income to Illinois as a Resident, but get credit for tax you paid to New York.
You will probably break even with both states, but since New York may have higher tax rates than Illinois, may get a bit back from Illinois. You are mainly just being a good citizen and reporting correctly.
Here's more info on filing a Non-Resident return:
https://ttlc.intuit.com/replies/4776808