The Kindle Fire version 1 charging port was never installed properly on the Kindle as it wasn't soldered to the board at the two points where it should have been. The broken part simply fell out when the system board was exposed.
The new part was loosely in place at first but in the proper location. I plugged in the charging cord and positioned the port so that the charging light came on. (In some positions it didn't.) The 5 little "feet" of the charging port have to touch the system board for this to happen--but then they can't move once charging begins. Thus I discovered the position that now needed to be anchored. I then soldered the two anchor holes for the port on the other side of the board where there had been no solder before and where the positioning pins of the charging port poke through the system board.
The Kindle Fire now charges properly and is back to normal use. Hopefully your repair will be successful.