Not sure what's going on with Kmix?
I use both headphones and speakers on my computer. I'd love to be able to switch between them using a keyboard shortcut rather than:

Click on volume
Settings > Audio setup
Prefer headphones
Apply

That is likely to wake my roommate up!

So is saving an audio profile supposed to allow me to change such things quickly? Or what does that do?
Or can I pass a command to kmix?

Thanks!

Like, oh my SQL!!!!

If your system uses Pulseaudio then you might be able to script this using it's tools. The "Audio Setup" UI is actually completely separate to KMix - it just makes it easily accessible to make things convenient.

"pactl" is probably your friend here - I believe KDE uses the "module-device-manager" module to control where audio goes.