Stephen
Even with the picture is not easy to see whats needed for the repair.
On the basis that at some time after you have fixed it the handle will need to be removed when access is needed inside the door, the repair should be capable of being dismantled.
If you drill and tap the shaft and put in a decent thread (M5 or M6 say) you could file the outer part of the shaft to it has a slight taper. If you do this with a coarse file you should be able to end up with what is in effect a tapered spline. Then put a slightly tapered hole in say a brass or aluminium bush which you can attach your handle to. Using a cap head bolt you can securely lock the two parts together.
If the shaft is steel and the bush is some softer metal the drive will be very strong, the two tapers dint have to be perfect, the rigged surface and the bolt tension will combine resources.
As others have said, a trip to the scrapyard might be easier.