Cut the branch roughly the size of the broken handle or to a size that is more comfortable to you and your hand.
Then when you are happy with that, drill a hole in the handle at the center at one end.  I used a drill bit that was smaller than the steel it was to be placed on, so that it would be a tight fit, as is seen in Image 3.

You can sand the branch, round off the end, varnish or paint it. I just left mine as it is. I prefer the natural shape, look and grip that the bark provides.