what you need is the good old "monkey glue and patch kit" that we used to use in the
70's when i was a kid.
first things first, you need to remove the tire from the rim using either a set of tire removal levers, or the ends of 2 crescent wrenches. never use a screwdriver or anything that might put more holes in the tube while you try to take the tire off the rim.
In the "monkey glue and patch kit" you would get rubber patches of all sizes. the patches had a sticky side covered with plastic that you had to remove prior to use, and a non sticky side. some rubber cement that was VERY flammable and a metal "cheese grater" type inner tube buffing tool.(this was actually the lid of the can, and the can was made of thick cardboard.)
once you have the tube out, fill it with air and stick it in a bucket of water. this will help you find the hole(s) a lot easier.
first you scratch up the area on the tube where the hole is with the "cheese grater" type buffer. this creates "teeth" in the rubber innertube for the cement to adhere to.
you smear some cement on the tube and let it tack up for a few minutes. remove the plastic on the patch and put the sticky side of the patch against the glue. put the tube on the ground and then roll the can that the patch kit came in over the patched area to get the patch nice & flat. this also helps squeeze out the excess cement. then you light it on fire, which burns the excess glue off, and heats up the glue under the patch and permanently bonds the inner tube rubber with the rubber cement.
seriously, this is how we did it back then and this is exactly what the instructions told you to do. it worked every time and we rarely got a hole or leak in the same area after we did this patching method.
when you put the tube back into the tire, make sure that the tire is half mounted ot the rim (one bead on the rim, one bead off the rim so you have a void between the tire and rim to stuff the tube back into the tire. first, put the valve core in the tire and line it up with the valve core hole in the rim...then put the valve cap on it. this will keep the valve core from pulling back up into the tire while you stuff the rest of the tube into the tire.
get the entire tube into the tire and remount the tire to the rim. 9 times out of 10, you can re-mount the tire to the rim without any tools...just use your hands to pry the tire over the rim. use a little bit of soapy water, spit, or glass cleaner on the rim. this will help get the last 2-3 inches of tire onto the rim a lot easier.
if the valve copre ends up crooked in the hole int he rim, now is the time to straighen it out. grab the valve core with your fingers, and rotate the tire keeping the valve core still. rotating the tire will turn the tube inside the rim so you can get the valve core straight.
when you fill the tire with air, squeeze the tire with your fingers above the valve core while you hook up the pumpand while you pump air into the tire. this will prevent the valve core from being pushed up inside the rim.
the problem is that you probably can not find the old type of rubber cemenet that came with these kits anymore because it is very toxic, flammable, and the fumes are poisonous when it is burning.
i'm not sure if the "new" tire patch glue will even burn, but if it will, i would try this method with it and see what happens.