HDMI output means that the sound is sent via the HDMI cable, when you connect your Laptop to a TV for example. I doubt that you have any HDMI cable connected to your PC, so that's why it says not plugged in. I'd say, that has nothing to do with your problem.
Anyway, did you ever plug any external speakers to your computer and got sound? Or is this the first time.
Btw it shows some sort of VU bars next to the speakers section in your Sound window? What happens when you play music? Does something move there?