I don't think flashing the bios would resolve your issue. I think your problem lies within the hardware itself, Does you laptop experience any extremes? Like prolonged periods or heat, or any rapid temperature change (which may lead to condensation), or possibly could it have been exposed to a surge somehow (Though this last is unlikely given your PSU is still in good shape)? My guess is the sound section of your motherboard is damaged somehow, and if that is the case, you may be SOL. I believe this for two reasons, one being because the internal speakers do not work, and only part of the external speakers work.
Before calling it quits though, since you are able to open and investigate your laptop, make sure the jack you used is connected correctly. After-all "Correlation does not imply causation. "
My specializations lie outside the realm of laptop specific motherboards, but from the sounds of your problem that is my guess.
Who knows though, perhaps somebody else has an Idea I am overlooking.