I'd try to see if you know anyone willing to take the darling in. I'm personally very wary of shelters. The one closest to me has a 58% kill rate which is pretty dismal. One of the cats I own actually belonged to a friend of mine who's roommates gave it to the shelter claiming she was a stray. She was put immediately on death row and was deemed unadoptable because she was shy. I saved her demanding 'my cat back' and have had her for three years now. She's one of the sweetest cats I've ever met in my life. Honestly... unadoptable.
If you keep feeding the stray the chances are he'll stick around if he starts feeling a bit better, hopefully long enough for you to figure out what to do.