Wow...that really doesn't look good... There may even be some damage inside the plug rather than how it connects to the board. You may need to find a replacement plug... If you want to try and fix that one, I'd probably try and do is gently rub off some of the burnt area with some emery cloth, but be VERY careful not to scratch the board or any nearby components. Then add a very small amount of solder to the top (what you pictured) three connectors and the bottom.
While the case is open, you may want to plug it up and see if you can see it wiggle any. Don't force it too much and break the solder, but confirm you did reseat everything.