OK, as previous responses ask, "What law is being broken?" At best she is following existing KY statues by not issuing a license for a same sex couple. At worst, she is following the SCOTUS ruling by not issuing ANY marriage licenses, because it ruled that KY's existing law is unconstitutional. Either way, this is a clear case of a Christian being targeted by homosexual activists and federal judicial overreach. Trump's answer clearly shows he doesn't see it that way.
Judges have the legal power to hold people in contempt.
In the Christian side, Jesus has the victim of trials that had false witnesses and false charges. He subjected himself to authority because that authority was placed over Him by God. This woman’s authority, the judge is also placed over he by God.