Your Denon is fine for playing the karaoke music but it was never designed to sing through. You could damage the Denon and any AV speakers it's connect to.
You should consider a powered PA speaker to sing through.
That warning said, to make this work you need a HDMI audio extractor like this:
https://www.amazon.com/gp/product/B019T0WTV4
It would go in between the Chromecast and the TV.
You would take the audio out of the extractor into the mixer along with the mics. The output of the mixer would connect to the Denon.
Your speakers are OK for the music but not designed to sing through. You could damage them. The same with your receiver. I realize your not interested in a separate PA for singing through but that's what I recommend. Your setup is good for the music but not for singing through.
If you were an audio expert and familiar with the concept of gain staging and the operation of a limiter you might be able to experiment with the settings to set it up properly. But I know these thing and even I wouldn't take the risk of damaging your speakers from someone else screaming into the mic or accidentally dropping one. That's all it might take to blow your speakers.
The Shure model number is for the receiver only. I'm assuming you have a couple wireless mics to go with them as well.
IMO the JBL KX200 is way too complicated for home karaoke use. That's more for a professional install where it's more set it and forget it. All the settings require drilling down through menus and submenus. Instead of the KX200 I recommend a small mixer with physical knobs like the Mackie profxv3. But there are digital mixers you can control wirelessly through a phone or tablet. The Behringer XR-12 or Soundcraft Ui12 are pro level mixers that won't break the bank.
Your audio extractor is probably OK. Try changing the audio settings on your Firestick to PCM and see if that fixes the problem. I've used this audio extractor just fine but I was going to a PA https://smile.amazon.com/gp/product/B019T0WTV4
But if it still doesn't work using the optical output on the TV works well https://smile.amazon.com/PROZOR-Digital-Converter-Optical-Toslink/dp/B00KNNSKV0