The lan socket isn't powered, there's no power there.
Inside the connector, there's a tiny transformer which isolates all pins in the ethernet jack from the computer, but that's it. The signal comes from (and incoming data is sent to) a network card chip which is circled in the picture below :
In rare cases it can happen for someone to accidentally break the chip or damage some components around it when inserting a video card into the pci-e slot. if a person is not careful the metal bracket can push down on components there, or can short circuit pins on the chip as the card is placed in position to be screwed to the case.
Visually inspect the ethernet jack and make sure the pins inside don't overlap one another (all should be parallel and sort of springy when touched) then make sure the onboard network card is enabled in bios - you pulled out the bios battery which may have reset the settings and therefore the onboard network may be disabled by default - if it doesn't work, I guess you're out of luck. It's really unlikely that the ethernet jack is faulty, it's higher chance you damaged something on the board. In that case, easiest solution is to just go in bios and disable the onboard network, then go and order a pci-e x1 network card, they're cheap, you can find some from 2$ and up (on ebay), and stores like newegg have some for around 10$.