Speakers don't work but headphones work

The bottom one, under Sound, video, and game controllers.
The other one (two) will disappear along with it until you install the driver.

It might be interesting/helpful, before you do the clean driver install, to check and see if there are duplicate or phantom sound devices showing up in your system.

Just go into Device Manager and go to the View Menu, the check "Show Hidden Devices".
If there are duplicate sound devices you will see them now.

Let us know what you find.

So there is no obvious double audio device that would point towards a driver conflict, but the multiple instances of I/O devices might.

Completely removing the device and all previous drivers, then install the new driver clean will clear that up if it is the case here.

OK. Sorry to hear that didn't work.

Let's try to isolate the problem to either hardware (defective laptop), or software (a Windows problem).
Find and download a copy of Ubuntu (or any other Linux program). You can save the file to a USB stick, then boot from the stick into Ubuntu Linux without installing it on your hard drive or affecting your Windows installation in any way.

Once in Ubuntu, see if your audio works.
If it works in Ubuntu then we know it is a Windows or driver problem.
If you have the same problem in another OS, then you will know there is something wrong with the PC.