It is seldom feasible to repair an axle shaft that has failed. If it failed it wasn't strong enough in the first place, and welding it will make it weaker, not stronger. Most axle shafts are heat treated, and after welding the heat treatment is gone, and the repaired area is more or less normalised.
You need a new or used axle shaft. I think a new one is a good idea unless you find out what killed the old one, and can be sure it won't happen again.
