It's hilarious how I come to this forum and people are talking about hens crowing like it's the most normal thing in the world.
However, I think I have a solution for you, and it does NOT involve killing her. Sheesh, it seems like that's some people's solution to everything.
I have several roosters, and in the winter at least, they stay close to the house, in the greenhouse. Unfortunately, they are right under the window of a certain person who sometimes wakes up when they start crowing at about 4 in the morning, or during the night. So, if it's bothering too much, we'll put them in a cage and take them to a warm place for the night, like the van (don't want their combs to freeze off!). Or sometimes, we will put them under the table in the greenhouse in a cage for the night. You have to make SURE to let them out though as soon as you can. When it's warmer out, we simply let them sleep in the chicken house, which is far enough away from the house. Maybe you could put the hen in a cage for the night, then take her to a safe place (e.g. the house), or have her and maybe even the other chickens sleep in an area that's farther from the neighbors.
Someone here had a design for an Eggloo chicken house that automatically opened at a certain time. You would still have to be really careful with something like that to make sure the hen doesn't get left in for too long.
In the mean time, I would recommend giving your neighbors a few spare eggs to keep them happy. And who knows, even if you don't do anything with the hen, they may just get used to her. The 'sensitive' person in this house has totally gotten used to it by now and doesn't wake up, so we don't even bother to put them somewhere out of earshot anymore. It's too cold anyways!
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Actually, some hens DO crow, as crazy as it seems, especially if there's no rooster around.