Electrifying a cage for rodent problem
Such electric traps are commercially available, but they're rather more complicated than connecting a small battery to a cage. My brother had a problem with rats, bought one of these, and we fixed it when it stopped working, which is how I know what one solution is. I'll go into enough detail so that you can see the sort of thing that's involved.
To kill humanely, you need enough current and energy across the heart. This particular trap used a narrow diameter plastic tube to exclude larger mammals, with two electrodes on the floor. A 500 μF photo-flash capacitor was charged to about 320 V by a standard photo-flash inverter circuit, but the potential was not applied to the electrodes. Why this complication? I would hazard a guess that there would be sufficient of an electric field that the sensitive rat would sense that something was not right, and not go across to the second electrode. A small sense voltage (&lt;1 V) was therefore placed across the electrodes, and a thyristor fired the cap into the circuit when a resistance across them was detected. Needless to say the protection circuitry needed for the sense amplifier was interesting.
You'd be wasting your time with any lower voltage, or a continuously applied higher voltage. I suggest you buy commercially.