Most of these jacks short out the internal jack pieces to allow for the speakers to work. In other words, inserting a headphone plug will interrupt the normal flow to the speaker and send it to the headphones. That being said, along with your description of what is happening here, I would suspect a bad connection on the board for the jack itself. If the solder connection is bad and loses connection to the board, it will appear to the system that a plug was inserted. That poor connection would also explain the requirement for an exact position for the plug as well. When in that particular position, the connection to the board is made.
I would open up the unit and look at the connections on the board at the jack. If you can, take a picture and post it here as well. Hopefully a simple resolder of the joint is all that will be required.
Dan