I don't think you will be able to be rid of them completely. You might be successful in catching the current ones, but others will move in. We have mice and voles living outside here and our rabbits and chickens also live outside. We are very careful not to leave any food around, which you say you are doing. We do still feed birds via feeders about 5 foot off the ground.
I am actually quite relaxed about them being in the garden. I see them as being no different to the other wildlife which we experience here. They have never made nests inside either the chickens' or the rabbits' sheds. If they did I would try to catch them in humane traps and release them outside.
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