Sue--to humanely kill something caught in a live trap:
1) Slip a double-loaf plastic bread bag (i.e., Costco size) inside a cloth sack to form a double-layered bag.
2) Fit it over one end of the trap and gather any excess so there's no room for your critter to squeeze through and make a break for it.
3) Feeling through the bag, lift up the little locking bar so you can open the door.
4) Push down the correct (this is crucial) lever to swing open the door at the bag end of the trap.
5) If it doesn't immediately run into the bag on its own, usually you can provide the needed encouragement by blowing a puff of air at it through the side of the trap.
6) Shut the door behind it, squeeze the neck of the bags shut, slip it free from the trap, and shake the critter down into the bottom.
7) Gripping the double-layer sack firmly by the gathered neck, swing it, fast and hard, in an overhead arc (like swinging a flail) and slam it onto a concrete sidewalk. The critter dies instantly. Remove plastic bag with dead critter inside, drop bag and all in the trash, get a new plastic bag, and reset your trap. Marking "notches" on top of the trap with a Sharpie is optional, but I like to keep track of how many rodents I've escorted to that big orchard in the sky (or, perhaps it's considerably further south so they can torment deceased fig-fanciers who falsified their E-bay offerings).
I started using the cloth outer bag when centrifugal force caused a particularly portly pack rat to rip through the bag at the zenith of my swing and wing its wide-eyed way towards the sun. A palo verde tree slowed its re-entry sufficiently to let it come to rest, disoriented but uninjured, in front of my garage, and doubting my ability coax it back into a new bag for a repeat performance, I opted for the pellet gun before it had a chance to regain its equilibrium and scamper for the high hills.