Since you are working in Florida for your own convenience, and you sometimes went to the New York office, you have to pay New York state tax on all of your wages. New York treats it as if you actually worked in New York State.
If you lived in a state that has state income tax, double taxation would be a concern. But since Florida has no state income tax, you don't have to worry about it.
It is indeed strange, and wrong, that your W-2 doesn't show any state wages. You can probably file your New York nonresident return by entering your W-2 with the information that should be in boxes 15, 16, and 17, even though it's not actually there. In box 15 select NY as the state. For the employer's state ID enter the federal tax ID, but without the hyphen. In box 16 enter the same amount that's in box 1. Leave box 17 empty or enter zero.
It looks like TurboTax will solve the New York City problem for you. Enter the local information in boxes 18 through 20 as they appear on your W-2. The city tax withheld will effectively be credited towards your state tax, because the New York return just adds the state and city withholding together. When the New York interview asks about your city residency, select Nonresident. You will get a screen pointing out that you have New York City withholding but you said you were not a resident of the city. Just click Continue.
Not having had any New York state tax withheld is going to cause you to have a big payment due to New York, plus a penalty for not having had enough withheld during the year. I assume that the city withholding is not nearly enough to cover the state tax. For this year, 2023, you need to tell your employer to start withholding New York state tax, stop reporting city wages, and stop withholding city tax. You might want to make estimated tax payments to New York, or temporarily increase the New York withholding, to make up for not having had any New York state tax withheld for the first half of this year.
Since your W-2 shows New York City wages, but your tax return will not have any New York City tax, you might get a notice from the city or the state saying that you have to pay city tax. You will have to explain that it was a mistake, and that you were a nonresident of the city for the entire year.
