A plain steel part like the flap lever should be repairable by welding, and is allowed per AC 43.13.
However, I do recommend that you remove the entire handle, have it blasted clean and checked for cracks thoroughly.
Then have it welded on a bench (instead of on the airplane), re-primed / painted and then re-installed.
The most important thing is that this is not a very common failure, so you should figure out why the handle weakened or cracked in the first place.
If you do decide to replace the handle with another one instead of repairing the broken one, please contact me before you throw out the broken handle because I would like to have it for a product display.