How do I keep a long haired rabbit clean?

I have a lionhead rabbit who is gorgeous. She is cleaned out every three days at the moment. For some reason she rarely cleans herself, particularly her feet and face and they are constantly dirty. I am already cleaning out her hutch more often than I was expecting but has anyone got any other ideas about how I can keep her clean. I have seen people with angoras put mesh in the bottom of the cage but it seems a little cruel for her to be walking around on wire.