The fact that you have no category for "Sound, Video and Game Controllers" in your device manager seems to be the main problem here.
I believe the items you are looking at under "System Devices" are actually Windows components, related to, but not the same as the actual sound device hardware.
So the question becomes: why doesn't your system detect any sound devices?
The first place to check is in your BIOS/UEFI settings. See if the sound device is disabled there.
Also it would be good to know if your system has HDMI video capability, as HDMI has it's own sound device that (should) show up separately.
Also, go to Control Panel > Sound and tell us what you find there.
But in any case you will not be able to install any sound driver until the system recognizes a sound device.