I like to think that despite of her disability she lived her best life, being surrounded by her siblings, enjoying the sunshine and eating fresh grass and bugs.
It sounds like you gave her the best life she could've possibly had, but were also aware enough to know that she wasn't likely going to enjoy it as long as the others.This is why even though I use the broomstick method I go ahead and give it a final yank to remove the head, to "make sure." Very little resistance since the spine is already disconnected at that point and it bleeds very little, for those who are squeamish about that.I prefer to use axe as I've read so many horror stories here about botched killings and knowing that the head is gone gives me certainty of the death.
