If you are in north Queens, I'd say to try TM&T. Looks like a hole in the wall station, just west of the Harley Davidson store on the north side of Northern Blvd around 50th? street. Across from the old Pathmark.
They're tire guys and surprisingly good tire guys who have been there for two generations now. If it is something like the valve stem being out, or the tire being off the bead, odds are they'll fix it without ripping you off.
There's only so many reasons that a tire "won't inflate" and anyplace that thinks it is rocket science, should be targeted for a rocket attack.
These odd spares MUST be available unused from the junk yards, after all there is one in each smashed Q5 and not much else uses them, does it?
Herr Doktor Strangelove: A brilliant product of German engineering that rides on four wheels and has just a few, heheh, eccentricities.