Most successful route in curing a small mice problem? Cat? Traps? What are your views?

hello. Im thinking of adopting two cats so they can prevent mice from entering our home. ( I will love and take care of these cute hunters) However, since mice carry diseases---I'm a little worried for the cats. As far as traps---I have a few in the home but it wont help because the mice will just renter the home in some way or another. ( trees ( I have yet to trim) , holes ( I have yet to seal) My house is always clean and all my food is sealed. Plus, I have three dogs ( so no poison unless I confine them) I just dont know what route to take. Please help!!! I called terminex and they charged me $200 to place 11 traps ( what a rip off) We checked the attic-- no droppings or signs of mice....