I think you did just fine. It's hands down the worst part of keeping critters, but you did what was best for your girl regardless of how hard it was on you and that takes real courage.
Dora'smom :
The broomstick method is what I used to help my poor Twiggy out of her misery. You turn the bird upside down, and take any broom or garden implement with a long handle, and lay it across the neck, putting one foot on the "stick" part on either side of the bird's neck. You continue...