It is probably the circuit board in the wiper motor. There are probably some broken solder joints that need repaired. There are only three screws to remove the cover, but one of them is near impossible to get out because of it's proximity to the engine (valve cover if I remember right). Patience and a really short torx bit with 1/4" wrench, it took about 10 mins to get out.
A word of caution though, when you take the cover off, the circuit board comes with it. There is a rotating lever in the center that is contacted by part of the wiper linkage as it moves back and forth. If this doesn't get back in the right place, the wipers won't work correctly. Unfortunately, when I put mine back together, I didn't take any pictures of the right way because I didn't want to take it back off again.
When you are jiggling the connector, you are moving the circuit board, that is probably the connection that is broken, connector to circuit board. A few touches with a soldering iron and you should be back in business.
