My take on this, would be to take the broke threaded piece inside the rod ball end.
Then chuck the cylinder rod on the lathe, face off to clean the broken remaining. Then drill and tap for the same size thread it had and use a piece of all thread, or cut the head of a bolt off or machine a short threaded stub. Finally, would simply thread the ball end to the rod, probably with some sort of thread locker. Hopefully I explained this clearly.
But I do have a lathe though. It's not technically necessary for this repair but does make things easier.