There has been some discussion of techniques to sail with a failed rudder, but I didn't see any comprehensive and simple explanations. As I just experienced a non-practice event (I think the day this thread was started), I thought I could share what I learned (and remembered eventually from basic sailing, and professional aeronautical knowledge).
I was out for a sail, and was going to meet up with a colleague (in a 14' dingy) and it was a very nice sail day, fairly sunny and steady winds 15+ knots. I was reefed to about 2/3 sail just to take it a little easy but I felt my rudder (TI, 2012 year) wasn't fully down. I retracted, but it jammed, and I used a lot of force to redeploy, but it didn't make it fully down. Now a little off course, I applied full rudder, and the pin sheared right away. Unfortunately this was pin #2, as I found the upper clevis portion sheared at the end of last year (the lines held it in place then, but before realizing the problem, it just seemed less effective).
Now for the important part. After reefing and assessing things I first started out reefed to about 1/3 and used the paddle to steer, and home was upwind. I couldn't come close to holding course. Not having too far to go to reach the shore of the island home (in the St. Lawrence just W. of Montreal), I tried just pedaling and steering (no sail), but to a similar effect. Sailing down wind was easy enough, so worst case I could land on the opposite shore and get a taxi home (no worries for the rapids 12 mi down wind). So after nearly giving up in fighting the wind, I eventually discovered almost by accident, that by opening nearly fully, the aerodynamic stability to wind would be just slightly positive, and then I could steer a straight course pointing nearly as well as with the rudder. In hindsight I realized that I should have figured out much sooner how to achieve "weather helm", but nearly full sail in strong wind w/o a rudder seemed unwise.
In a conventional boat, reefing the main sail moves the center of pressure forward, but not nearly as dramatically as our roller reefing mast. So when significantly reefed, the sail TI/AI is stable on the leeward end of the boat so it wants to go downwind (lee helm), and when full, has the opposite tendency as I previously described. I didn't try to see if I could hold a beam reach but that might make good practice another time.
I was back in the water on Monday after visiting the dealer and backing off the rudder bolt, and slightly shortening the up/down lines too. The aft position up-rudder pulley was damaged (the line jumped and jammed in the block) so we swapped with the front position until the replacement (and a spare) arrives.