For ease of use I would use this mixer. Not quite as many knobs but a little more expensive

https://www.amazon.com/Mackie-ProFX-Mixer-Unpowered-6-channel/dp/B07XC3BHTL

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For ease of use I'd go with wired microphones instead of wireless.

https://www.amazon.com/Shure-SM48-LC-Dynamic-Microphone-Cardioid/dp/B0002D0HY4?th=1

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Some of the cables would change. If you're interested I'll look those up later.

The mixer is a good choice.

The cable will work from your laptop to the mixer if your laptop has a 3.5mm output

Connecting the laptop to the TV via HDMI allows you to have a separate screen on the laptop to operate Karafun while projecting just the words to the TV

There are many wired mics out there that will work fine for home use. But I recommend the Shure SM48 with a on/off switch

https://www.amazon.com/Shure-SM48-LC-Dynamic-Microphone-Cardioid/dp/B0002GZF5W

There are many powered speaker(s) that will work well for home use. I recommend this one for a low cost speaker that I personally own. I like the 10" woofer version because it's not too hard to move around.

https://smile.amazon.com/Rockville-RPG10BT-V2-Bluetooth-Wireless/dp/B088ZH5CGH

if you want something a little more substantial you might consider this

https://smile.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ