Just a couple of amendments to what Stefan has said.
Pit bike wheels come standard with 15 mm spindles. They are a bit on the light side, 17 mm works just fine (Refer to Chris Brown bearing chart)
Honda wheels as standard are 12 or 15 mm. The same applies. Again, check that OD of the bearings, I have had some with 32mm OD bearings, Learn from my costly lack of knowledge!
If using Gemini stub axles, both varieties are 17 mm (Honda or Pitbike), the difference is in the length, at least the ones they sold me on each occasion that was the difference.
I have used plain steel on the 10 mm kingpin, also used bronze, and also kart spindles with bearings and 8 mm king ins. All have worked OK for me. These cyclekarts are not a fit and forget, just make it so you can change things readily along the way.
David F-R, your method sounds like an engineering masterclass, I suspect that very few would feel the need to go as exotic as that.