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We had ten eggs successfully hatch.  Number eleven started to hatch but he stopped making progress.  He cracked his shell and started to work his way around it but then stopped.  He stayed like this for about 2 days.
We could hear him chirp and make microscopic movements but for some reason he wasn't going to make it out of the shell.  At about midnight last night we checked on the chicks before heading to bed.  We had read that it can be dangerous to help a chick out of their egg but we couldn't just let him stay like that because he wouldn't survive.  If he was trying to hatch then he had probably already absorbed all of the nutrients from yolk of the egg so he would starve to death soon.  So Stuart decided he had to help the chick get out of the egg.  Here's the video of his chick extrication.  It's almost 10 minutes long but even then we weren't done freeing him.
Although the chick was free from the egg, he had a layer of dried stickiness that we hadn't seen with the other chicks.  It is almost like the yoke had broken, leaked onto his head, and then dried.  It had not only glued a piece of shell to the back of his neck but it was also causing his head to stick to his body so he couldn't lift his head at all.  No wonder the poor little guy couldn't turn around in his egg to finish breaking out.  He was glued together and couldn't move.
Since this chick was in his egg longer than he should have been, he had made his first poop inside the egg.  It was green and messy but we had to get him cleaned up.  Baby chicks can't regulate their body temperature and they need to be kept warm so we couldn't just give the little guy a full soaking in a bath.  Stuart patiently worked on him with a damp Q-tip.
Once he was cleaned up, I used a hair dryer on a low setting while Stuart cradled him in his hands.
We were hoping to dry him off to warm him up but also to fluff up some of his feathers.  Unfortunately, his feathers seems to have to same weird coating on them and they just wouldn't fluff up even though they were dry.
So here is the little guy after working on him for over an hour.  He still had the piece of egg shell on him and his head was still stuck to him.
I love seeing Stuart's tender heart in action.  He was so sweet and gentle with this tiny little chick.  We just weren't sure what else to do to help him.
We decided to use a little olive oil on a Q-tip to try to help clean away the dried on stickiness.  After about 20 minutes, Stuart was able to get the goop cleared away enough that the chick's head was loose and he could move freely.
We took a break for a bit and gave the chick some water with a dropper.  I'm sure he was feeling 100% better than he had been but he still had that piece of egg shell glued to him.
Stuart worked on his some more with the olive oil on a Q-tip.  It was very time consuming and it was already very late at night but Stuart kept working on him.
Finally he was able to remove the egg shell from the back of the chick's neck.  We used the hair dryer a little more but there was still some dried goop on his feathers and now there was also olive oil so we couldn't get the feathers to fluff up.  That could be a problem for him keeping himself warm but it was now 2:00 in the morning and we had to get to bed.  We tucked baby chick Eleven back into the incubator to stay warm for the night.