You're either going to have to start playing your music through a player like VLC (which can actually load YouTube videos with some work) or use an entirely different device for your music.
For example, when I stream I use Spotify on my phone to play music. That phone is connected to my keyboard's mic input via a male to male 3.5mm cord. I then tell windows to "listen" to that device on my headset.
For game audio, I have my on board audio set as the default device, then have Stero Mix set to also listen on my headset.
Finally I'm using a virtual audio cable to grab Discord/TeamSpeak and also listen to that on headset.
All in all when streaming, I have 4 channels; game, music, chat, and mic. All of which I can control the levels and balance as I see fit. I can even make it so I myself don't hear the music but my stream does, which can come in handy while playing competitive in CSGO.