A while back I "Robin Hooded" an arrow with a self knock. --well, it wasn't at the bulls eye, and it wasn't intentional, so it doesn't really count-- but the knock was broken either way, and on one of my best arrows. Here is the process I have been using to fix broken knocks on a self knock arrow:
1. Find a shaft of the same diameter or sand one down to match the broken shaft. (first picture)
2. Measure the new shaft to the length needed to replace the broken portion and add about 3/8". (picture 2)
3. Next create the new knock. I have a jig that lets me set the depth and center the hole in one move. I drill out the end and then cut down to the hole and use a file to widen the slot. (third picture)
4. Carefully cut off the broken knock below the damage. Make sure you keep the cut square. I then drill out the arrow shaft 3/8" to accept the plug. Make sure the hole is straight and square to the end. (fourth picture)