Thank you! I had this problem for 3 years now and finally had the time to work on this thanks to your post. I was having a hard time getting the servo arm back into the socket (at first it seem like the arm is too short to reached the socket). Then after 30 minutes of frustration, I stumbled on how to get the arm and socket close enough to snap it back into place.
It was not clear to me from OP post that the internal guts of the light can pivot back and forth. Look at my photo and gently grap the big black wire and pull it toward the opening to bring the pivot closer to the servo arm. See my white circled item. Dont yank on it or you might break something. Just give it a pull toward the opening and you will understand what I mean.
iphone 4s camera so handy!