Have you tried a pair of headphones or a different set of speakers in the headphone jack? If headphones work in the jack and you get the "Headphones" listing in the Sound panel, then the problem may be with your speakers or the speaker plug rather than the computer.
Another possibility -- could "Built-In Output" you're seeing be referring to the headphone jack? I think your computer has only one audio out, so it might not have separate listings for "Headphones" and "Line Out." If that's the case, maybe click on that listing to see what the setting of the output volume is. Maybe it's muted?
If the problem ends up being the headphone jack on the computer itself and you're leery of attempting a repair, an alternative might be to get a USB to Mini adapter like the iMic. But I'd highly recommend confirming that your speakers work with another machine first.