Turning a ball on lathe
If the ball end is removable it will be easier to make a new one from case hardening steel and heat treat it after machining.
Turn the ball oversize by hand and with a bit of practice this is not too difficult.
Finish turn the ball using a hardened silver steel tool in the form of a tube with a sharpened end, which looks like a wadding punch. A diameter of about half the diameter of the ball is about right.
Mount this tool in the tool post and move it to engage the ball to be cut always pointing it towards the centre of the ball.
This produces a very accurate ball which can be finished off with emery cloth.