You're mixing a lot of symptoms and potential problems.
1. Your laptop probably has a built-in mic.
If your headset is in fact not being recognized, this is probably what's being recognized by Steam/Discord.
2. If in fact "Windows can't detect your headset" then nothing can see it, so it can't be going to Steam/Discord, something else, (1) is at play there.
3. You didn't say what kind of headset.
If it's USB, things are different.
If it's a TRRS/smartphone style set of earbuds/headphones with an inline mic, things are different.
You're jumping to the assumption that this is some sort of driver problem, but without more specific information, we can't begin to figure out what's going on.
If it's a 3.5mm TRRS headset that you're plugging into the headphone/headset jack on your laptop that uses the onboard audio, try some other headphones. If that doesn't work either, and you otherwise have audio in Windows, your physical jack is probably busted.
If it's a USB headset, which it doesn't sound like, try a different port as suggested above. Otherwise, probably broken.