This is a known issue with the iPhone 5G. In my experience it is one of two things, either the battery has a poor connection (or sometimes even a dead battery) or the main-board is a goner.

Open the laptop as per instructions from the UX31A keyboard guide. Make sure the ribbon cable that spans the CPU fan is properly seated on both ends. If this does not work, the replacement board is separate from the motherboard and is usually called the "IO board" or "media board".

Try using a hot glue gun. The glue is too thick and dries too fast to spread into the actual connection. I also doubt it'll melt any of the cable sheathing, but just in case, heat up a drop, wait a sec or two and see if you can mold it into the joint with your fingers.