Do you take the view that all possible remedies have been explored and there is no point in trying anything further.? Is installing a sound card a DIY procedure for a low-to-medium tech: !!? Regard, Alpha. :-?
Yes, I'm inclined to think continued efforts to fix your onboard sound are going to be fruitless and installing a sound card is the solution. If you've never done any hardware upgrades, this could be somewhat challenging. I suggest you take a look at this reference and decide whether you want to try installing a sound card:
Illustrated How to Install a Sound Card Replacement. Note that the illustration actually shows a second card, a "daughter" card, being installed along with the basic sound card; you won't need to do that, just install the sound card, and then install drivers.
If you've never opened your computer before and handled any components in it, you should be aware that avoiding any discharge of static electricity from your body to those parts is very important. You can do so by simply touching the metal chasis, i.e. unpainted interior case frame before touching any computer components [motherboard, memory sticks, any add-on cards (sound, video, modem, etc.)]. Most sound cards will ship with instructions on installation but may assume you have some knowledge of installing components and, therefore, may not show the type of illustrations in the above link.
Concerning this forum discussion, I believe a couple of basic questions have never been asked. Have you checked your speaker connection to verify your speakers are properly connected to your computer? Do your speakers have their own power cord? Is it plugged in? Have you checked your sound control panel to make sure Mute is not enabled?