Maybe you fried the audio card accidentally while replacing the SSD?

Does HDMI audio out work? Or a USB soundcard?

Sounds like you might have shorted out the analog portion while working inside your case. You mentioned other digital methods (like HDMI) work. If you have a connector to a front jack, reversing the pins can fry the audio (been there done that). It’s also possible something accidentally touched and shorted it out.

Most likely you’re gonna end up having to get a USB or other audio card.

Try the obvious stuff first, in case you missed anything. It's easy to do :)

Reboot the computer. Check that everything is plugged in properly, whether it's your speakers or headset. Check that Windows is trying to use the correct device. It may have switched the default to your fixed monitor, especially as you get sound through it. Check the volume for your headset. Windows supports separate volume levels for different devices. Check the headset in the rear connectors on your computer. Check your headset with the cable on a known working device, like your phone. Boot to a live Linux distro and see if the sound works there. Turn the computer off completely, and make sure that you didn't unplug the front panel connector when you installed the SSD.

These should give you a good idea of where the problem is, and let you rule out a few things. Obviously you don't need to carry out the steps that would be redundant, so if your headset works with the rear ports, you don't need to check it with your phone, for example.

Good luck :)

What do you mean by fixed your monitor? I ask because windows likes to make the monitor speakers as the default output when plugging it into the computer.

Boot a Linux cd and try it in that, rule our software first.