It is hard to say which approach to take for machining.
I have good luck using square die drills and machining using a ceramic insert.
In my experience you DON'T want to weld an extension on. The old shaft is high carbon steel and at will go very brittle due to welding.
It is usually easier to make a coupling.
If that is not possible you will need to weld then anneal and then re-harden and likely straighten it.