Why are the mice there? Because momma mouse has found it a good place to raise a family. They can find shelter about anywhere but there is food there. Even if you store chicken feed in a mouse-proof container like a metal garbage can and take the feeders up every night, they are still finding food there. Chickens will eat mice so they have to be careful, but they will come out and eat during the day. Plus when your chickens are feeding, they spill some feed on the coop or run floor. That will attract mice. It is extremely difficult to get rid of mice or keep them away.
I also consider poison the last resort, but you have been given some bad information. A chicken will not die from eating one poisoned mouse. It’s about dosage. A mouse weights somewhere around one ounce. A chicken big enough to eat a mouse weighs a few pounds, many times bigger than a mouse. The amount of poison that kills one mouse won’t kill a healthy chicken. I still don’t like poison. It’s possible a chicken can eat more than one poisoned mouse and the effects can be cumulative. Poison is an indiscriminate killer. It’s possible your chickens or other pets can get to it. A rat might carry it out of the secure area to an open area. I had a rat move the block of poison once, but that was far from my chickens and still secure from my pets. Still, it can happen. I don’t recommend poison unless you are really desperate, and I don’t see anything in your post that sounds desperate.
Without using poison I don’t think you will ever totally get rid of them, plus even if you do, more will just move in. It’s not about totally absolutely getting rid of every mouse forever but you should try to keep the numbers under control. They are nasty creatures, pooping everywhere they go. They can spread some diseases, especially when the numbers get out of control. They can do a lot of damage with their tunneling and chewing. Mice will attract snakes. Some of us don’t mind snakes around, especially non-poisonous ones, but some people do. Snakes can also eat eggs and small chickens, how small depends on the size of the snake. I think there are lots of good reasons to not want mice around.
So what can you do? Try to keep the area cleaned up, don’t pile up trash and keep weeds cleared. That will reduce the number of safe places for them but they can still hide a lot of places. Still it will help. Cats or some dogs can possibly be real good at reducing the mouse population.
I recommend you start a trapping program. There are a lot of different traps out there designed with mice in mind, sticky traps, live traps, snap traps, and some homemade ones. Which one works best for you depends on your unique circumstances. You don’t want your chickens to get to them. Pets could be another problem. You don’t want them injured or for them to spring the trap. If I use a snap trap, I wire it to something solid so a raccoon, possum, or cat can’t carry it away and I try to put it where a larger animal can’t get to it anyway, for example. How are you going to dispose of the mice you catch? I like to feed mine to the chickens. The mice need to be dead for that and not poisoned.
I don’t know the best way for you to approach it, that will depend on what you are willing to do and your unique circumstances, but I do recommend you try to keep the mice numbers down. Good luck!