Your employer does have a method they use to determine the amount they need to withhold for New York (NY) while you were working there. You just need to use that number in your W2 for state tax withheld in NY. This is actually what they paid to the state of NY on your behalf. It would be like your federal withholding, in that, it is an estimated tax amount that may be required based on the earnings for the period.
You may or may not owe additional tax, likewise you may receive a refund for overpayment. Once you allocate the days as indicated in the previous answer, you will see the actual tax liability and refund or balance due for NY.
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