Make headphone output mono
This is another situation where a program like JACK, or possibly Virtual Audio Cable, can come in handy.
JACK works like a recording studio patch bay, allowing you to take audio input from one source or program and reroute it through other programs for processing before getting to audio hardware.
For your usage, you'd use JACK to route audio through some type of mixer plugin (VST or similar) that will downmix the stereo output to mono before sending it to the soundcard. Optionally, you might duplicate the mono signal to both left and right channels so the soundcard thinks it's a stereo signal.
I've never tried Virtual Audio Cable, but saw it recommended on another question and it sounds like it does basically the same thing as JACK.
I've noticed on a few occasions that I seemed to get mono sound on both sides when I accidentally pulled the plug partway out of the headphone jack.
Some video players have options to play only the left channel or only the right channel on both speakers. For example, in VLC Media Player, you can click on Audio, Audio Channels, Right.
(I think that would be better than looking for a Mono setting. A Mono setting will mix the left and right channels together and send the mono mix to both speakers. In your video, that might cause the music to be louder than the voice.)