You didn’t mention the type of trap you are using. I assume it is a standard snap trap, and if the mouse is alive, then it has probably been caught between the shoulders and the hips.
You can reduce the likelihood of this event in a few ways. First, always be sure you set the snap trap so the narrow side of the bait end is abutting the wall or other vertical surface. This will encourage the mouse to be struck in the neck area by the bar. See Mouse control for images on proper placement or search the eXtension site. Second, you can purchase quick kill traps. They are more expensive but they are designed to force the mouse to approach the trap in a certain way so that the strike bar (which is usually beefed up) has a significantly increased chance of a quick kill.
Euthanasia is defined as rapid loss of unconsciousness, followed by irreversible cessation of heart and brain function.
The best overview of these principles can be found at:
Euthanasia PrinciplesRemember, the bottom line with any form of euthanasia is, “Does it result in rapid loss of unconsciousness, followed by irreversible cessation of heart and brain function?”
As far as deer mice and CO2, there are procedures and guidelines (e.g., Euthanasia of Rodents. I doubt you will use this same procedure at home. For ideas on how to build your own CO2 euthanasia chamber visit Euthanasia Chamber.
Y Cervical dislocation is probably quickest and most efficient in these cases (see Mouse Cervical Dislocation, since you can control the mouse by handling the trap and the tail.
If you have captured the mouse in a box or cage trap, you can place a plastic bag over the opening, drop the mouse into it, twist the bag down to control the animal’s movements, firmly grasp the mouse behind the head, and apply cervical dislocation or decapitation Cervical Dislocation. Or you can release it far from your home! Be aware, however, that survival of translocated mice is uncertain and some states ban the translocation of wildlife.