The KY in box 15 just means that the employer was willing to withhold taxes for the state of Kentucky (are they in Kentucky?) - I imagine that they print KY on every W-2 whether they withhold KY taxes or not.
On the other hand, no number in box 17 does NOT mean that you don't owe any taxes to Kentucky - it just means that you didn't authorize your employer to withhold any.
Yes, this likely means that you owe the entire tax due amount, since nothing was withheld. You need to talk to your employer about Kentucky withholding.
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