While the "core" for the audio solution is provided in the chipset, a separate CODEC IC completes the solution -- and it is the driver for this CODEC that you require. This driver is provided by the IC vendor and passed through to you by the board/system vendor (in your case Dell).
I looked at Dell's support page for the Optiplex 780. This is an absolutely ancient Intel Core 2 Duo/Quad-based system. Full and proper support for Windows 10 is not provided for systems this old. Hardware-specific drivers are not provided. You can try and find an Audio driver for your system (it has an Analog Devices ADI 198x family CODEC), but I doubt you will find one. I recommend you return this system to a Windows O/S for which drivers are actually provided (in this case, Windows 7).
P.S. You are likely mad after reading this. Microsoft had no business telling people to upgrade systems this old to Windows 10 - and worse, in many cases, forcing it to install.
This message was posted on behalf of Intel Corporation
The information provided by Scott is correct.
Please check the list of supported operating system for your computer model.
See page 7
You can spend a mere US$7 and get a reasonable audio solution (see here: Amazon.com: Sabrent USB External Stereo Sound Adapter). I bought a couple of these for use in emergencies and have been happy with their performance. If you want a higher quality solution, there are some excellent high-end USB and PCIe-based audio cards out there that are fully supported by Windows 10.
Hope this helps,
P.S. Let's put this in perspective: Your board is approaching being 7 board generations and 7+ years out of date. That is not my definition of "becomes redundant so quickly". Many of the components on your board are approaching or past their (typically 7 year) MTBF dates.