While replacing the handle like Craig suggested is likely the best route to go, there is a glue I use for practically everything broken.
It’s called ‘Goop’. It comes in automotive or marine applications (in a squeeze type tube)
Being waterproof, it’s ideal for outdoor use. (actually better as this stuff gives off a very powerful odour until it dries)
I used it on a car I used to have where the moulding around the corner of the bumper was coming off.
I used alchohol to clean both areas and applied a substantial amount to each, waited a minute for the glue to get tacky then pushed the moulding into place, kept pressure on for a few minutes until it held.
Two years later it was still stuck on tight even after multiple drive through car washes.
Since then I’ve used this glue for all types of applications. (there are no restrictions as to the type of materials to be glued)
IF you want to try it on the door handle (***mind you, there are NO guarantees in this application):
First, (center) drill a 1/8" hole in both sections about 1/2" in depth.
(You need to make a pin now)
Second, cut a nail (I think a 2" finishing nail is of the right diameter) 1" long. (No head)
Apply glue into each drilled hole and the face of each section, insert the pin, align the handle and hold the 2 sections together until they stay.
Leave alone for about 24 hours so the glue really sets hard before use.
Remember, this glue needs fresh air ventilation, so work safely.
***Caution, this glue is flammable.
I’m not sure where you are located or which store you have that carries ‘Goop’, but up here in Ontario, Canada, it’s available at Canadian Tire.
I would try stores like Walmart or perhaps automotive stores.