Lets get some more diagnostic info on whats happening. Download this great app: CoconutBattery it will allow you to see the status of your battery and the charging logic. Take a screen snapshot of the main window and paste it here for us to see Adding images to an existing question
You should also review this Apple T/N: Apple Portables: Troubleshooting MagSafe adapters making sure both the adapters connector and systems are clean and in good shape.
Well your battery is getting up there with 763 cycles. If you review this Apple T/N: Determine battery cycle count for Mac notebooks we can see your battery is rated for 1000 cycles. So in three or so years you've run your system hard. You may want to plug in more often not letting the battery run all the way down. When you do it often causes the battery to blister.
The good news is SMC appears to be working so its only the switching MOSFET that needs replacing on your logic board. As the battery is not switching over this is likely from the water damage.
You'll need to find someone with the micro-soldering skills and access to the part to fix this.
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