The best way to re-attach your faucet handle with the broken tack-welds is to remove the socket from the shaft, and either weld it properly or braze/silver-solder it properly. Be sure to keep the correct alignment so you can access the set-screw through the handle-hole.
The more likely solution if you don't know how to, or are not equipped for, welding or brazing stainless steel is to clean it up as much as possible and then epoxy it into place. Be sure not to fill the inside of the socket with epoxy, or glue the set-screw in place. A "putty-type" epoxy may be better for this than a liquid/runny type. Be sure to give adequate cure time for full strength.
You can, of course, hire the welding done, but it may not make economic sense. If you know someone who will do it for a 6-pack, it might. Most commercial welding companies will price it sky-high because they don't really cater to consumer welding needs.
However, it is also likely as mentioned in comments that your new faucet is on the way, despite you not understanding how they can do that without asking for your address.