So, last May, I went out and used all that money I saved up to buy myself a new desktop to replace my old semi-functional dinosaur of a laptop. It was long overdue, as my laptop was from way back (2006) and I might've been tempting fate by attempting another year on the thing...

Anyway, all was reasonably well until today. I turned my CPU on this afternoon and discovered that my computer had no sound, neither with headphones or via my flatscreen's admittedly weak speakers. Just a red "x" visible where I would usually manipulate the sound in the lower right hand corner. Clicking this brought up a delightful message informing me that my peripheral was not plugged in.

Now I should elaborate that it is not my TV (my PS4 producing all the expected noises on screen when hooked to the same TV), and my headphones are not at fault either. Headphones were completely connected.

A virus scan uncovered nothing of much relevance, and I agonized around this for quite some time before finding a way to access my list of drivers. I could be mistaken (because my tech capabilities are admittedly lacking), but it appeared that I suddenly had no audio driver.

I went to the Dell site to download new drivers, and after getting all the way to the end of the download, it very politely informed me that the download failed.

I was all like, wtf, but right as I was ready to throw my hands up and haul my damn Dell out the window nearest me, I tried plugging my headphone directly into the jack located on the computer itself...and found it actually worked!! Nevertheless, audio still does not come from my TV, but only when the jack is directly connected to the computer itself, and I am at an utter loss as to explain why.

Does anyone know what could be the problem here?

Try system restore. If not try the Dell forums, others may have had the same problem.

my experience with anything Dell has not been good. i worked for dell when they upgraded the nashville facility. they used their own computers to operate the whole place, they could not keep them up and running. that facility shut down every other facility worldwide when their system crashed. it took almost a week to fix that mess.
if you have "upgraded" to windows 10 fret not they will have a work around available in a few weeks (maybe). you're lucky "10" didn't turn your pc into a brick as it has to MANY.
my recommendation is switch to Linux. the Mint edition is very windows like. it can be run from a thumb drive to test it and get used to the terminology then if you want install it to your hard drive alongside windows. they do not interact in any way.
my nephew is running "Ubuntu" on a dinosaur and is really satisfied with the way it functions. i just started using "Linux Lite" on my HP laptop. i have had no problems with it at all. it is faster and has a smaller footprint and very light on resources. my harddrive hardly comes on as opposed to windows, which made it run constantly. with the "Lite" version i have access to all of the "Ubuntu" repositorys and forums. i don't know if "Mint" has the same access. i am very happy with "Lite" and am 95% sure windows will be coming off this machine in the very near future. i think you would be happy to make the switch. LOTS of peopl are making the switch. oh i almost forget Linux is Free!