If you just need to shower you can get a day pass to a gym like NYSC or the YMCA:
NYSC
YMCA

Many gyms offer free trial passes but you would have to pretend to be local and sit through a membership sales pitch which would be tiresome after a long flight.

For someplace to nap, the Y has reasonable rates for spartan rooms.
Or head to the public library and fall asleep w an open book on your lap :-)

You could also try a day admission (~$35) to an area bathhouse / spa
Russian
Wall Street Bath House
Korean
Spa Castle in Queens NY
King Sauna in Palisades NJ

The Korean spas are clean with plenty of nap areas but are not in Manhattan.
Spa Castle is ~20min cab ride from LGA, further east from the city.
Both spas have free shuttle service from designated pickup points.