The County Clerk of Rowan County refused to follow a judicial order and the County Clerk of Rowan County is being held in contempt for refusing to follow the judicial order. Kim Davis and her supporters believe this is about her - the individual. They are wrong. This is about the position, which happens to be held by her.
If Kim Davis wants to get out of jail, then she can do one of two things. She can issue the marriage licenses the holder of the Office of the County Clerk is obliged to issue, or she can resign, and allow another person hold that office and be bound by the obligations attached thereto. She doesn't get to have both, and this is where she's stuck.