At last a question I really am qualified to answer. My parents bought a caning and rushing business in High Wycombe when I was 6, so I've been surrounded by chair repairs all my life. A steel plate can work, but you'd be surprised by the forces exerted by both rush and cane.
Repairing a rushing rail with the rush in place is very awkward. Usually rails break because they rot, often at the joint with the leg. If that happens the most effective solution is to cramp up the chair as well as you can, drill through the leg and into the end of the rail and insert a length of dowel or steel rod with epoxy, then plug the hole.
Repairing a rail that's broken away from the ends inside rush you want to keep is more difficult. If it's a clean break you may be able to glue and screw it back together, but it's unlikely. Instead I would try to pull the two broken parts out of their holes (watch out for pins holding them in place, which need to be weedled out first). Use the broken rail as a template to make a new one. Fortunately rushing rails only have to be rough. Then cut the raill in half at the fattest point at an angle to create a scarf joint. Drill pilot and clearance holes across the joint. Feed the ends back in, with glue on the ends after a trial, then glue and screw the joint, but don't use PU glue which can foam everywhere. It can help to keep the chair rigid with a sash cramp and spreader cramp working in opposite directions.
Any questions email me, and my wife runs the business now and rush the chair for you if all else fails.
Good luck
Nick