I would agree with
@redflash that the quickest and easiest speedometer would be an app on a GPS enabled cell phone. I tried a few mounts and have not found one that work good enough to see while I am driving. Most car mounts are fragile or expensive and wil let the phone shake loose on bumps, so be aware your expensive phone could get damaged.
Now I use a pouch to hold my phone and protect it at the same time (hands free, but out of sight),,, it's a bummer when I put it there and the GPS app is asleep or not recording, but it's easy to get the GPS going and take another lap.
To use a bicycle speedometer, you will have a couple of mods to make.
The bike speedometer has a drive tab that sticks in between two of the spokes and turns the speedometer cable (?) time(s) per 1 tire rotation. The problem is that bike speedometers that I have seen mount on the end of the front axle.
My first thought would be to drive the speedometer at the base of the rear axle sprocket, but that may not be possible because the cable is supposed to mount on the end of an axle, and a bike axle is like 3/8" while a kart axle is 1" or more.
If you can figure out how to drive the speedometer, you will also need to re-calibrate the speedo head for accuracy because the speedometer is meant for a specific wheel diameter (like 24" or 26") and go-kart tires are smaller diameter, so the speed will appear faster because the speedometer iwill probably be overdriven.
Smaller tires = more revolutions for the same distance
One way to fix that is to re-draw the numbers on the face plate. You would have to use some math skills or measuring skills to figure out the % overdrive (% error) and re-draw the scale.
Best way to calibrate it would be to compare to a GPS app on your phone (ha ha)
Good luck.
Hands-free phone mount sounds better, right ?
I hope you can show some pictures of what you end up using.
I am showing a picture of a phone mount that fits in a CD player slot. I drilled holes and mounted it to my steering wheel bracket, but it eventually broke with the weight of my android phone on a yard track. For reference you may need something sturdier than this.
Ive been having good results with a tool pouch that I got at the hardware store. Just tie wrap it somewhere and slip in the phone or tools, etc.