Normally if it was just the display or backlight that failed then the external output should still function. I see from the specs (thanks for including the link) that it has both VGA and HDMI outputs. You didn't say which one you tried, but I would try both. If an external display doesn't work on either then I would say that the GPU has failed. In that case a motherboard replacement would generally be required as a repair. A quick search shows that may in fact be a common problem for these systems
, and one way to repair the motherboard might be to "reflow
Before going that far I would disassemble the system to disconnect and reconnect the internal display just to be sure that a bad connection isn't causing the problem. You might also try running an external monitor with the internal display unplugged to see if that makes a difference. If none of that helps, the only options I can see would be to try reflowing the existing board or to replace it instead. Let me know what you find, Scott.