A hole the size of a dime will admit a rat. I know you may not believe me but it's true. Rodents have ways of squeezing through things without breaking bones, go look it up. You need to block every single hole, get underneath your AS and you will find many. Some people use steel wool, some people use canned foam, all it needs to be is completely inedible or unusable by the rodents. Food is not the only thing that draws them, they love paper and cotton and other nesting materials. Never assume there is only one mouse, and when there are two or more, the little buggers ARE prolific like you cannot believe!
My husband and son refused to trap/kill the tiny cute little mousies in our apartment and soon, within a week, we were so over run that I'd be cooking and the little butts would be popping out of the top of the stove and running across the counter!!! I finally bought mousetraps and soon snap! snap! snap! was all we heard for a week. They called me a murderer, I was cruel and inhumane 'witch' as I threw the still wriggling little bodies into the trash, (happened twice). But after a full week with no snaps, I put away the traps. I bleached everything to death, and the house felt clean again. Then they admitted they liked it better without mice, but I was still a murdering witch.
Traps are generally the quickest way to kill a mouse. I don't approve of glue traps, or drowning. If I had to choose how to be killed, I'd want it to be an instant death, wouldn't you?
When you are cleaning up after mice, always wear gloves, damp spray, (very very damp), with a bleach solution, and maybe even wear a mask. I know that only certain mice and rats carry Hanta Virus, but why take the chance? Especially if there if maybe your mouse got in when you were camping. You get Hanta from touching or breathing in the dust from their droppings, body fluids and fur.
Directions for Proper Clean Up of Mouse/Rat Droppings, Nests, etc. Apparently I was not as thorough as I thought I was. So I'm a lucky gal considering how many cabins I have cleaned in my time as a Scout Leader.