I'm a little late to the game, but here's a couple of things:
Your HDMI / Display Port outputs will show up as Audio Devices in Device Manager, since they can carry audio. If you don't have anything plugged into them, they won't be available to select when you choose which audio output to use. (this has caused me problems when the audio gets sent to an HDMI monitor that doesn't have speakers, but I digress)
And Vekseid is correct. "System Speaker" isn't your soundcard, it's the simple beeper that's been part of PCs since the early 80's. Not all machines have it anymore, and my desktop (that does, I hooked it up) doesn't have the device installed in Windows.
If you did a system restore of windows and it fixed it but came back...it might be a hardware issue. My suggestions, check your BIOS / UEFI settings to make sure the onboard audio device hasn't been disabled, and is showing up there. In device manager, you can check the option to "Show Hidden Devices", to see if your sound card is there but not being shown. Then you can try to do a complete restore (back up your files first), and see if that helps. If not, you can try taking it somewhere, or get a pair of USB headphones. (Or bluetooth if the system has it).