/img/smilies/hugs.gif First, let me say I'm sorry for your sad situation.
In my opinion, breaking the neck is the most humane, it seems to kill the chicken instantly. I use a different method to dispatch the meat birds but with dear old friends I prefer to do it in a less messy, more dignified & respectful way.
Sometimes it's easier to have someone else less emotionally attatched to the bird do it for you. Then they can focus on doing a quick effective job and not be so personally saddened by its loss. Find a friend or relative who is willing, or find a neighboring chicken keeper with whom you can trade these sad favors.
I recently had to euthanize a hen who had become an internal layer, I owed it to her to end her suffering. I first dug the hole in a pretty part of my garden near a flowering shrub. Then I held her under my non-dominant left arm and calmed her with kind words & gentle stroking. Then I strengthened my resolve, grabbed a hold of her just behind her head with my stronger right arm, and gave a decisive firm yank, pulling out & twisting slightly. I knew she died as soon as the neck bones gave way with a crunch. Of course there were some reflexive flappings of her wings for a moment afterwards, it took about a minute for her to grow still & her eyes to close.
It's essential to give a really hard yank. Once you release the head & see the neck is hanging limp, you know you've done the job. You can continue to hold the bird, or lay her down, cover her with a feed sack or towel, & walk away if you cannot bear it. That's why I say sometimes it's easier for someone else to do you the favor, so they can concentrate on doing the job well for the bird's sake.
You can also lay the bird on the ground with her neck under a thick straight stick, like a broomstick. Hold the ends of the stick down firmly with your feet, grab the legs & pull up really hard until you feel that crunch.
This last time was the first time I tried doing it while holding the bird under my arm. It was both successful (for the hen's sake) and -- don't get offended by my choice of words -- more satisfying for me as her keeper, it felt nicer to hold her right up to her end instead of laying her on the ground to wonder at the last "what's going to happen to me now?"
I wish you peace with your decision & your actions to follow.