I got out a screw driver and opened up the case. The brown power supply board was easy to get to and check out. It's kind of difficult to see in the picture, but I immediately noticed a suspect capacitor with a bent up top. I plugged the TV up and turned it on and sure enough I noticed a high pitched hiss coming from around the capacitor.
At this point I was fairly certain that capacitor was the problem. Still, I decided to remove the board and give it a proper going over.
Since I had a spare capacitor of the same value conveniently lying around, I decided to go ahead and replace the one I suspected of being damaged. With a little solder wick I managed to remove the faulty component and replace it with a nice new blue capacitor.
After placing the power board back in the case I plugged the TV into the outlet and pressed the power button and sure enough it was working again! All that trouble from such a small component...