I checked the HP site for drivers for your system, and they only show 32 bit drivers. Also, the only audio driver is for a "Thin" speaker, which does not seem to be related.
If it has an onboard sound system, you may need to use the chipset drivers for it, but again, not x64 drives on the HP site. I checked the Intel site for your chipset, and it looks like the link might be the ones you need. But it is had to know for sure since I don't have all the info on the board, but I got these by using the chipset 82801 reference you supplied. So check yourself to see if they look right.
http://downloadcenter.intel.com/SearchResult.aspx?lang=eng&keyword="82801"
Chipset drivers are for most of the basic systems. If you decide to install these, I would make a backup or at least a good restore point, since problems might arise and you could be unable to boot the system. This probably won't happen, but you need to be aware.
As far as the Realtek, you may need to go to the Realtek site and download drivers from there. The best I could come up with is:
Link Removed - Invalid URL
I will not be able to provide a link farther along because there is an "I agree" checkbox that will need to be done. But these drivers would not seem to be for your HD audio, just the legacy type.
This seems to be an older box and sometimes drivers for older devices do not have the functionality you might need.