Can anyone recommend a reliable home health agency that serves the Long Island/Nassau County area?

Your best bet is to find a list on the Internet, call them and set up an interview.   Aging Care website goes worldwide, getting someone here from Long Island to give a recommendation might be slim.

As for live-in coverage to do 24 hour care, that might be difficult because one person doing all the work will burn out quickly.   It might be better to have 3 separate shifts of caregivers, that way when one's shift is over, they can go back to their family and get a good night sleep.   Also, it gives the patient more than just one person to talk to during the day :)   It's something to think about.

White Glove is one agency I worked with although not for homecare contracts. I am no longer with them, but they are a very large agency an may be able to accommodate your needs for home care.