This happens every now and then on pinball machines, as over time heat and vibration will take a toll on the game, and since the solder joint is (usually) strong and the wire is the weak point, the wire breaks off. The required fix is to re-solder the wire back to the post. Normally the wire is soldered to the underside of the post, but I decided to solder it to the top because it was easier and more visible, it really doesn't make a difference either way as long as there's no danger of shorting, which in this case will not be an issue:
It only took a couple minutes to remove the old solder (which still had the remainder of the wire mixed in), strip the wire back, and re-solder it to the post. I fired up the game and the ball kicked out perfectly. I love easy repairs!