Judge Bunning wisely placed Kim Davis into jail for contempt of court. She's the Kentucky county clerk that famously refused to issue marriage licenses to gay couples because of a religious objection. The judge correctly surmised that any fine would be covered by sympathetic allies.
Due to political considerations (read: cowardice) her elected position makes her "unfireable" and immune to removal from office. So now it's a standoff, she remains in jail until she agrees to perform the lawful duties of her office. In the meantime, marriage licenses are being issued now in her county.
I think there could be a possible solution here, but it could take a few months to work out. It appears that her term of office is two years. If that's true, the voters in her county could, if she is still in jail come the next election, simply vote her out of office. If she were not the county clerk, there'd be no reason for her to remain in prison.
But, the voters have been putting her in office time and time again, for many years, why should they vote her out? Many agree with her illegal position on gay marriage licenses.
Well, they would be doing her the favor of getting her out of jail, and taking the first step to letting the embarrassment of Davis' actions have had on their county begin to fade. If Davis doesn't fold up beforehand, this path could ultimately be the way out of the senseless stupidity that Davis' attempt at martyrdom and a book and movie sale has caused.
Of course, if she were to win the next election, well then it'll be interesting to see how the foolishness will ultimately be ended.