To the OP: What a shame, that you were put in the position of finding it necessary to kill a child's pet. I do sympathize with this. Not a pleasant choice. Obviously you should never have had to make this choice.
This thread is exactly why I would not post here about killing an animal, unless I were unsure of what choice I should make. Once we make the choice we feel is right, we do not need to read an onslaught of alternate opinions. Surely, few if any here would take an animal's life casually, without respect for life, or in fun, or out of spite, or for similarly, let us say, uncivilized motives. This is a chicken site, and there are members who are vegetarians, and others who raise chickens for meat. Obviously there are varying viewpoints. We need to remember that a wide variation of opinion or viewpoint can still encompass thoughtful and moral approaches.
OK, off my soapbox now.