Chosen Solution
Transferred everything from the old, and twisted, back housing, to the new back housing. Nothing happens when I try to charge and turn on. I am assuming that it is some kind of grounding issue but I don’t know where to look.
You mentioned the old housing was twisted. If it was, the logic board could have conformed to that shape too. Putting it into a normal housing could have forced it back into place, which put tension on the board, which could have damaged the board. Take out the logic board, and examine it for any damage. When reconstructing, try and make sure all the cables are properly in place. Then, see if it is detected by iTunes when plugged into your computer. If the computer doesn’t recognise it, I would then take it to a microsoldering technician.