If you have a digital multimeter, you may be reading induced an voltage (80V) and there isn't any current on the wire. A small lightbulb might be better to test with. You should get 120V between hot and ground and hot and neutral and zero volts between ground and neutral.
Personally, I've never seen K&T wiring with a ground.
You'll need to map out the whole K&T run and deduce where things are going. It's usually pretty tough, especially with three way lights and fly-away splices leaving different places and meeting up somewhere else. After you have a map, then you can troubleshoot.
Personally, I'd use this oppurtunity to upgrade the wiring to meet todays code.
I do not know the proper way to splice nm/thhn to the ruibber coated K&T, but I would think it would be wire nuts and not just tape. K&T to itself is actually soldered and taped, not just taped.