Help with a punctured tire

The tube itself is not holding air?  Take the whole thing apart.  Bring the tube to the kitchen sink or the bath tub.  Pump some air into the tube, just a few pounds, put it into some water in the sink and watch for bubbles to tell you where it's leaking.  If it's a tiny hole, you can patch it as you previously did.  If this nail caused a pretty large hole, I'd replace the tube.  And if the gash in the tire is as bad as it sounds, youmight want to think about replacing that as well.  It may be pretty weakend in the area where the nail was, and could cause a sudden blow out, something you don't want to experience at 30 mph.

if the hole is in the tred, and when you pulled the nail out, the hole is clean, then the tire is fine.  If it is in the tred and when you pulled out the nail, you see pieces of fiber sticking out (inside or outside) then the tire is toast.  If the hole in the sidewall, the tire is toast.

I have never had luck with a patch kit either.  They always leak on me, and the patch does go on the tube, not the tire.  Think of it like this;  the tube hold pressure and that's what supports the bike.  The tube is too thin and would easily puncture, so they put a tire around it that will take the abuse.  As long as the tire's structure is sound and the tube can't bubble through it, then the tire doesn't have to be anywhere close to air tight.

It sounds like you are comfortable changing a tube, so just go buy a new one for a few bucks and while you are in the shop ask show someone the tire and ask if it's trashed or not.