Cut two strips of paper as wide as the key and 1/4" longer than the part remaining on the handle. Mix up a tiny dab of epoxy. Put the glue on the end of the piece you have, then if possible, slide the pieces of paper alongside the part in the lock cylinder. It helps if the ends are folded away from each other a little to make inserting the handle easier. Slide the handle part in until the glue hits the piece that's stuck, then use tape or a rubber band to hold it there until the glue cures. The paper will prevent the glue from squishing sideways and sticking to the walls of the cylinder. You DO have to worry about the glue pushing up and down where it can stick to the tumblers, so just use a very small amount; just enough to get the job done. Give it a good day to fully cure and reach its full strength before you pull it out.
An alternative to the pieces of paper is to coat the outer surface of the glue with a light grease. Coat it all the way around except on the end that you want to stick to the broken part. If the glue squeezes out, the grease will reduce its ability to stick to stuff you don't want it to.