I have some trapping tips below, but if your only interested in the cat solution, skip to the end.
How many traps do you have set? If you are catching 1or2 mice a day then you don't have enough traps. 30 traps are better. I know it sounds excessive, but that is the recommedation for pest control experts. You want to have more traps out than mice that you are trying to catch. More traps are better anyway, because male mice give off hormones that can make small female mice afraid to approach that area. In other words: if you catch an big male mouse on a trap, you may stop being able to catch small female mice on that trap.
Wear gloves when setting traps to prevent the smell of your hands from transfering.
Mice are more suspicous of new objects and smells than rats. If you are using snap traps give at least a week for the mice to feel comfortable with the traps before setting them (you should still bait them during this time). It would be awful to have a mouse approach a scary trap for the first time and then get spooked by the trap going off. That mouse will not approach that trap again. You want all your mice to be comfortable with the traps, its their fastfood restaurant that they visit daily. Then one day you set all the traps and those mice won't think twice about jumping on them. If using sticky traps, everything I just said doesn't apply.
Also, make sure there are no bushes or hiding spots near the pen. Mice don't like running into open clearings, they prefer to run along walls or under dense bushes. You want them to feel exposed/unsafe. If you have bushes near the pen that you are unwilling to part with, you can raise the skirts 8inches up off the ground to make movement under the bush visible from a distance. This will make them feel unsafe, and will help their predators see them.
I have had cats and chickens, as has my neighbor.
These are my observations about cats and chickens:
Old hens will peck/pick on kittens. If you have territorial chickens, they may be a problem with small kittens that cant get away. However, aggression towards kittens is less likely with docile breeds.
If you have a rooster, cats will generally avoid spooking your chickens for fear of the rooster.
Large cats (especially male cats that haven't been fixed) may kill small chickens. If your cat is prone to killing large animals like rabbits or squirrels, then it may also consider killing a chicken. However, most cats are more likely to go after small twitter birds, lizards, gophers, rats and mice. Most cats would only be a problem around chicks.
Cats that are raised around chickens normally learn to respect their space and leave them alone, and same goes for the chickens.
There are exceptional cats that love spending time with chickens. My neighor's cat, who was raised with chickens, loves to come over to visit my chickens. He will go smell them and roll over and expose his tummy. If they try to peck his nose, he will leave them alone. Some chickens find him very interesting and follow him around. Other find him frightening and run away (he then chases for fun).
Other potential problems to consider:
-Not all cats are good moucers. Cats with tall ears, big tails and long legs appear (in my experience) to be more interested in hunting than those with the reverse.
-Cats are nocturnal (this means that they are active at the same time as coyotes). If you have coyotes in your area you may keep losing your cats to predation. I have noticed that black cats seem to survive better in that situation.
-If you plan to keep your cat and chickens in the same coop then you will have to clean up any dead carcass parts the cat leaves behind/vomits up. Chickens will go after carcasses, and this can spread disease. Also, cats dig holes and bury their poop, so you will not be able to easily remove it. Chickens dig to find food. Though cats are not commonly known to transmit chicken diseases, the potential transmission of pathogens like toxoplasma gondi would be much higher in this situation.