I have a Sharp Aquos LC-40E77UN with the same issue, and have tracked down a single burned capacitor at C7816 on the power supply board, and am looking for info on the capacitor in order to replace it. (Power supply: RDENCA354WJQZ / Delta DPS 208BPA). - Any help with a schematic would be appreciated!)
Since you say the picture went out, it is possible that you might have the same problem, Bad/failed/leaking capacitors are, unfortunately, quite common in electronics. There are volumes of info available on the subject.
Replacement power boards can also be purchased. One for my TV runs about $60 to $120 depending on used or new. Since I was able to see a good picture on mine for a few seconds, despite the burnt out capacitor, I'm opting to try and find/replace the cap myself (using the proper safety procedures), or at worst, have a repair done to the power supply.
Backlights should typically last around 60,000 hours.