We have a similar issue. Our neighbors live like pigs. But we get rats and baby rats---not mice. ewwww!
Neighbors leave dog food and water out all day so rats have a buffet all day and all night. They have a gazebo thing that is falling apart but won't take it down or fix it. We think the rats live under/in the gazebo.
Our cat pretty much takes care of the rodents and keeps them from getting near the house. Ever thought of adopting a nice "mouser" from the pound/ animal shelter?
I would encourage you to NOT use poisons. They seep down into the the ground and down into the water supply. No one knows how long they last and stick around, and they are very harmful to kids and pets.
Mouse traps or those sticky traps are very effective. Still, think about adopting a cat and giving it a second chance. It'll have those mice running.