One of my hens startted eating eggs, and I finally caught one (and
marked her), now I would like to know the proper way to kill a
chicken. Any ideas?

One way I've heard that's not been mentioned yet (the wrist
flick being the best, quickest way if you just want to get rid
of it):

Hang the chicken by its feet over a bucket. Using a straight
razor or similarly sharp implement, slit the entire beak
lengthwise and let it drain into the bucket...