I'm very sorry to hear this but in the end its what nature intended. Naturally, the sick animals, predator and prey alike would likely be picked off before they get to the point of pain but as our pets are not likely to be preyed upon as a sick or injured animal would in the wild so they rely on us to do what nature would otherwise "clean up". Just remember, its as nature intended. Do what you feel best in your heart and you will never go wrong.
Keep your head up!
If you need to talk, feel free to PM me