It’s a rare karaoke machine that outputs a picture and audio on HDMI, which one do you have?

I’m not sure I understand how the output of your machine works. You talk about two inputs for the microphone and the karaoke machine, which doesn’t make much sense, Sonos is concerned, ultimately, about outputs, first to the TV, then from the TV to the Arc via HDMI ARC or eARC. 

If you supply a link to the manual for this device, we might be able to give you better advice.

In general, since most Karaoke machines output audio via RCA, it makes Sonos particularly unsuitable for Karaoke, due to the delay on analog inputs on all Sonos devices.

I can’t find the manual for the karaoke machine, but it’s Nakamichi NKX One. I use Shure SLX-D for the microphone. I’m thinking about putting both the NKX and SLX to TV input then to Arc via HDMI eArc. 

Interesting. I can’t find a manual on this device, but here is a sales page:

https://www.legatomusiccenter.com/shop/home-audio/karaoke/karaoke-media-players/nakamichi-nkx-one

It suggests the audio is sent out of the digital optical output, not the HDMI, although it really isn’t clear. I’d be interested if the singing and music you have sent out is played through the TV, and ultimately to your Sonos, or if the video is just the printed lyrics sent through the HDMI.  

Its feasible, if the audio is carried only via the analog out, it won’t work for you, but you could conceivably connect the optical output on this device to the optical adapter from Sonos, which would then feed the Sonos…but you’d need to disconnect the Sonos from your TV.

Not really wild about any of this, but it can’t hurt to try.