Where I am not a lot of people have money to buy multimeters so they are not a common household item... and even fewer people that know how to use them. I had a very confused electrician looking at it. Fortunately I didn't trust what he said.
I did figure it out though. On the top of the circuit board, the ground is marked with 'gnd'. I didn't see it at first because the wire is crammed against the outer edge of the aluminum cooling shield, among other things, and it was blocking the view. I am not sure why there is a + on the same connector below the board though. But I soldered it and its working
If anyone has the same problem with this model of charger:
On the circuit board there are two wires along the leading edge. On mine they are both wrapped in white shrink wrap leading off the board. They both enter into a bigger white wire in the cord. The center pin is the small white wire that enters the bigger white wire in the cord. The other connects to the middle connector of the barrel.
Along the edge of the circuit board there is a black shrinked wrapped wire. This is the negative that goes to the outer barrel of the connector. In the cord it is the outmost wire that wraps the white wire with the two other wires inside.