First load 'Computer Management' and go to the Device Manager tab. You should be able to find the sound card under 'Sound, video and game controllers' and the headset and microphone under 'Audio inputs and outputs' or also under 'Sound, video and game controllers'.
If they aren't there then it would confirm that it is likely a driver issue, you may well find you have 'Unknown Devices' listed, some of which may relate to your soundcard chip. If you can't get a Windows 10 driver for the VIA chip you have then the only solution is, as Greyghost suggested, to buy a cheap secondary sound card.
According to your manual the sound chip is a VIA VT1708s
NOT a Realtek. I found a download here:
https://www.techspot.com/drivers/driver/file/information/17949/ WARNING - I know nothing about this site or the content of its files. I did download them and they looked like what I would expect but it doesn't mean they aren't riddled with viruses or will work with your PC.
If the devices are listed then let us know.
Another common issue is that the later builds of Windows 10 disables the microphone in Privacy Settings by default. Search for Privacy Settings then go in the microphone section and enable it for Desktop Apps (there is also a separate setting for Microsoft Store Apps that comes first - don't confuse the two). However, this is only worth looking at once you have the drivers sorted.