Well, I'm back to answer my own question. If anyone else ends up in this thread trying to accomplish the same thing, here's what I did.
For $16, I picked up this
Bluetooth Receiver on Amazon. I then plugged that into the mic input on the PC. Make sure that you go to the Recording tab in the Sound control panel, double click on the icon for the "mic" you just plugged in. Go to the Listen tab and tick the box to "Listen to this device." There are some other settings in the other tabs that you can tweak to make sure Alexa isn't too loud or too quiet. Click OK and you're good to go.
I really feel like there should be a software solution to make Alexa see the computer as a speaker, but I couldn't find it. For less than $20, this problem is solved and I can get on with life.