You are correct. The school district number is not located on your W2, but is available on this list:
Ohio Public School District Numbers. However, since you are a
nonresident (which you can see in the opening parragraph when the form opens up), you will enter 9999.
Now, if you were a full-year resident of KY I know you are filling out an Ohio return, which you only need to do if Ohio taxes were taken out. KY and Ohio have a reciprocal agreement, that states you are only taxed in Kentucky because that is where you live. If Ohio tax was taken out, you will file a nonresident reciprocal return to get your taxes back. Be aware, however, that your income is taxable to Kentucky, so that is why you will need the taxes back.
On the flip side, if no Ohio tax was taken out, you do not need to file an Ohio return, just Kentucky. And if Ohio state tax is withheld, you can have your employer take out tax only to Kentucky by filing the correct form with them. The following FAQ provides more information:
https://ttlc.intuit.com/replies/4776822
Ohio local taxes will continue to be withheld, however. That doesn't change with the reciprocal agreement.
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