Your best bet is probably putting a knife through the vent in the roof of the mouth (hang her upside down). Speaking from personal experience, I strangled a chicken and I cried for hours over it; poor thing had broken her leg, I didn't have anything sharp enough to cut her throat (we live too close to neighbors to risk stray bullets) so I did what was advised in my Storey's guide to raising chickens. Place her on the ground with a broomstick over her neck and pull her legs quickly while standing on said broomstick to break her neck. I ended up strangling her because I didn't have the heart to pull her legs due to one of them being broken. I'll never forget watching her die and crying and telling her it was for her own good and that I loved her.
The next day i bought a butchering set.