I'm confused, because you seem to have answered your own question.
You know you need an input, as you comment the motherboard analog inputs are in use.
So you would cable the two machines together how? Add a compatible input on the PC and it's trivial. No available inputs on the PC and it's impossible.
So I don't understand your question perhaps. You could stream from the pi to the pc using dlna or something of course, over the network. But you want to use the pi's audio out, which means you need someplace to plug that cable in. What else is there?
Are you asking how to add another audio input to the PC? You could buy a soundblaster audio card and install it in an internal slot, or you could buy a USB audio adapter. Those can be had for cheap, and don't require opening the case. Just make sure you buy one that has inputs and not just outputs.