Reddit mentions of HDMI Audio Extractor Splitter HDMI in to HDMI Out + Optical Audio TOSLINK SPDIF+ Digital Analog RCA L/R Stereo Audio Splitter Converter Adapter Support 4K@30Hz 1080P Full HD 3D by DotStone

Not OP but I have a bunch of these hooked up and I’ve never had any issues with them. You don’t have to pay much :/.

I have some hooked up to an hdmi audio extractor with an RCA -> AUX cable into my PC’s Microphone port, which I have set to ‘Listen To’ in Windows 10, so any audio coming from the HDMI source gets routed through my PC. $35 vs ~$150 for OPs capture card method :P. Though, if your motherboard or sound card support SPDIF input, you should use that instead (this extractor does both).

Please do! Everyone deserves such a cozy den.

The one pictured is the "IC200B PICO Projector" (latest here).

Pros: What you see pictured.
Cons: The fan is a bit loud, I'm going to 3d print a noise shield/holster for it to hold it in place and deflect the fan sound back behind the bed frame. Oh and the annoying built-in speaker, I picked up some HDMI audio splitter ('mirrors' audio signal to audio output type of your choosing) to deal with it. I go chromecast > HDMI audio splitter > projector for video, then chromecast > HDMI audio splitter > speaker amp for audio. It doesn't remove the audio from the HDMI signal so a tiny bit still gets to the projector, but I can easily jack the audio amp up and it drowns out any/all sound from the projector.

I doubt you have a sound card; you're probably using the motherboard audio, and yes, it sounds bad. So bad.

Does your monitor have speakers or an audio jack? The HDMI to your monitor carries audio. If the monitor doesn't have any way to play audio, you can use an HDMI audio extractor; that might sound better.

https://smile.amazon.com/dp/B01KYURMEW/

You might be asking for this:
https://www.amazon.com/DotStone-Extractor-Splitter-Optical-Converter/dp/B01KYURMEW/ref=sr_1_1?s=electronics&ie=UTF8&qid=1491533298&sr=1-1-spons&keywords=hdmi+audio+extractor&psc=1

Notice it has Hdmi output as well.

Im not an AV guy, but I would guess you cannot play on two hdmi ports from your switcher, even if it had two outputs.

edit: please note i do not endorse this product or its counterparts in anyway, it just seems like the solution OP is looking for.

You can use an audio extractor, or find a way to directly connect your speakers to your monitor.

HDMI Optical Converter Amazon

I know it was kind of a jerk answer but he wasn't wrong about Googling it. First result was this one on Amazon and the great reviews, many say works as a solution to their PS4. My Astro headphones need an optical too (likely what you have), they plug in via USB for chat and accept the optical as the surround sound audio. Seems like it should work for your case.

PS: this is why I opted to upgrade to the Pro, needed the optical port.

There are dozens of different boxes you can get off of Amazon that extract the HDMI audio to optical, this one for example has good reviews:

https://www.amazon.com/DotStone-Extractor-Splitter-Optical-Converter/dp/B01KYURMEW/

DotStone HDMI Audio Extractor Splitter HDMI to HDMI + Optical Audio TOSLINK SPDIF +Digital Analog RCA L / R Stereo Audio Splitter Converter(HDMI Input,HDMI+ Toslink/Digital Analog Audio Output) https://www.amazon.com/dp/B01KYURMEW/ref=cm_sw_r_cp_apa_hXJkzbFB9SD3S

or buy an audio extractor like one of these https://www.amazon.com/DotStone-Extractor-Splitter-Optical-Converter/dp/B01KYURMEW/ which just gives the added benefit of being able to listen with the tv off.