Hotel near JFK

I'm sure this is a question that's been answered many times before, yet a search resolved nothing.

I'm on the BA180 next week and, having never taken the early flight before, think that a hotel as near as possible to the departure gate is my preferred sleep-maximising solution.

Google is being coy about the options - there doesn't seem to be anything at or in T7 itself, but there are a few Holiday Inn Express places just a spit away.

Any and all suggestions welcomed!

Thanks, OT