iPhone 5 with faulty logic board, repair?

This is almost certainly a failure of the U2 chip on the board. There was once someone on here who said he could do it. It may help to guide you to google "iPhone 5 U2 replacement service" or something like that.

The logic board has leak current problem. Try restore to the latest software to rule out software issues. If nothing changes, check all functions like camera or audio. Also plug it onto a Mac with Xcode to check device logs. You may be able to narrow down the faulty hardware like camera or audio chip not responding.

If everything seems to be working, then this repair is going to be difficult because it is very difficult to identify a broken component if everything seems to be working. Try starting with U2 as jessa suggested. U2 is the USB interface chip if I remember correctly.

If you replace the U2 chip and the phone works, blame Bono for the fault.