Short answer, No. If one has to ask this question, he's not an experienced home shop welder and machinist, who could invest no-charge hours in such a project. If a machine shop were asked to weld and turn true, they'd say "Fuggediboudit."
Most likely solution is a used part off eBay or Marketplace. Make certain the seller guarantees his shaft runs true. There are occasionally random junk parts being sold there also.
Also, carefully examine the cartridge and the housing for cracks and damage. A blow sufficient to break off the spindle nose is a bad one.
There are also #2 Morse taper spindles available. They can work, but will require an adapter to hold the chuck. The JT33 spindle which directly mounts the chuck is preferrable for home shop use.
Finally, removing the broken spindle nose from the chuck is difficult. I've drilled them out from the outside. I've also drilled a hole through the bottom of the chuck and punched them out from the inside.
jack vines
