Laptop Repair Guy points out that Toshiba has issued a warranty extension resulting from a class action lawsuit. As far as I can tell (I’m not a lawyer so better check for yourself) I’m out of luck since I’m in Canada but definitely better to have Toshiba fix it for free than try yourself.
As you see on the photo above, the positive connector oxidized and almost black. If you put a fresh solder on the connector like this, the power jack problem will reappear very soon. I usually remove the power jack from the system board (with iron gun and solder sucker), clean oxidized pins on the power jack and pads on the motherboard and only after that resolder the power jack.
It turns out, unsurprisingly, that he was right. The power jack started coming loose again after about a month and a half. I was in the middle of a bunch of things so I didn’t really feel like messing with it so I just opened up the computer and threw some more solder on it. When the power jack started jiggling again a few weeks later, I knew I had to do something better to fix it. So I picked up a desoldering sucker and some desoldering wick.