Again with reference to the article above, there is a possible mechanism -- impeachment -- but this comes with a couple of catches. Most notably it can be a rather drawn-out process, but also it may be relevant that, due to her being elected, she is ultimately responsible for conduct in office to the electorate. And in the middle of Kentucky you might well find more voters that support her position rather than oppose it.
In this sense it's not clear that there is any easy way of sacking her, as she could well argue that she's carrying out her job according to the wishes of her effective employers (the voters). But it is clear that there is a criminal case against her as she has refused to carry out her legal duties and rejected attempts to compel her to do so. Hence contempt of court (and imprisonment) is about the only avenue that could be easily pursued against her.
I agree that it is a fairly ridiculous position, and a sacking would be called for in other circumstances. But this case is highlighting how many problems emerge when multiple interests clash -- religious freedoms vs duties of employment, made worse by the arcane rules in operation in the US when it comes to elected officials. And worse still because she shouldn't have been elected at all.
In doing some background reading I discovered that the position of "property valuation" was an elected one as well! I just don't get the US obsession with elections all the time. And it throws up absurd scenarios like this one.