Wow, so much already said.
This happened to me, I posted about my experiences just a month ago. I just want to say that NONE of these solutions are perfect, everything depends on the wind/sea conditions, boat size/type, and your available equipment.
In my case, approximately 20mph winds and 4 foot seas deemed sail steering completely impossible, lashing anything undersized and under mounted was a complete waste of time, and if I hadn't had an outboard and a radio
to call in a PAN, I'd say it was moving from inconvenience to emergency pretty quickly. While the boat was safe from immediate danger the jibing, spinning, and uncontrolled drifting would've eventually been bad news inviting rigging
failure or other complications. Left to our own without an outboard and coast guard I'm sure we would've figured something out, but sail steering and lashing objects to the remaining rudder didn't work whatsoever and I have to believe any jerry-rigged drogue would've done similar.
I like the drogue idea, but I severely underestimated just how much force is required to keep a boat tracking straight in high-wind and seas... I have a few doubts about other people's suggestions. With a transom mounted rudder you can practice this sometime and see how it goes. With a hull/through hull rudder and wheel setup, just pray it never happens.
After my incident, I'm thinking some kind of backup rudder that is actually tested and capable is required gear.
I have heard of the wind-vane's rudder working as a backup.
Trust a guy who finally shook off the sound, you never want to hear fiberglass tearing and/or failing on your boat out at sea. It's the worst sound you've ever heard and you know you're not going to enjoy that sail for too much longer.