UPdate/Edit: Skip to the bottom, and if what I mention down there doesn't apply, come back and read the first part just below this paragraph: It dawned on me about 1/2hr ago that if the audio chip was damaged it probably wouldn't have shown up in Device Manager as working!
You have had the right idea, to upgrade that PC into one that will run games, but my limited research of the mobo tells me it wasn't the strongest M2N board made. The computer isn't crap, the HP's just aren't generally built "strong" enough for intensive gaming etc. IMHO.
Regarding the Device Manager: GREAT! the 'parts' show up in there and are supposedly working. The Nvidia stuff is for your HDMI audio, so I don't think we need to be concerned with this.
Have you gone into the BIOS and made certain that the on-board Realtek audio is selected? You are going to have to go in there when you get the new sound card - to disable on-board sound - so if you haven't tried that, give it a shot.
Re. a windows reinstall; yes, everything will be wiped out with a clean install of Windows; but make this the Truly last resort option because of all the reinstalls and saving of information you have to do.
Lastly, the question of parts failure. Unfortunately, sometimes things just wear out. There might have been a power fluctuation, a solder joint may have finally given up and un-soldered(in the chip or mobo itself), static electricity may have done it in: The list could go on but you get the point.
BTW, you have probably already done this but I went on ASUS website and cannot find this motherboard, even though I have seen pictures of it. Pegatron was a susidiary of ASUS until 2010, and susidiaries of Pegatron (since 2010) include ASRock . I could not find any information on their website re. the mobo, but it appears identical to the ASUS board.
That PC isn't ancient and you are on the right path to keep it going for awhile but at some point it will or has become more expensive, and frustrating, to keep running then it would be to build one, or take your psu and graphics card and buy a $400 pc at, say Wal-Mart, and install your upgraded components in the new inexpensive pc.
BTW, I unselected the solution so that others may jump in with more or different information. Thanks for selecting my answer and I would be pleased if you did so again, but let's see if we can fix the problem first.
Forgive me if you have already tried this.
Go to device manager>sound etc. controllers. Right click on the RealTEk Audio and uninstall it. Do Not restart yet.
Go to Programs and Features (Control Panel) and uninstall/remove everything re. audio by RealTek: Careful here, you have RealTek LAN for your internet and you do NOT want to uninstall those.
Re-boot and on reboot, go into the BIOS (press Delete or which ever key will get you into setup/bios, at the start of the power up). Find the audio selection and select on-board audio.
After re-boot, reinstall your realtek audio drivers. You probably will need to reboot again.
Then make sure RT shows up in device manager. It should.
Go to Control Panel > Sound and see if the RT shows up under Speakers. If it finally does, select as default.
If you have already done all of this, perhaps numerous times, call me a bad name and swear for wasting more time, then read the stuff at the top.
Have a safe trip, looking forward to some sort of reply (good or bad) on Tuesday.