The current state of the world is that marriage licenses are issued in Kim Davis' office by other employees. The licenses do not have her name on them. This is in compliance with an executive order issued by Governor Matt Bevin and later by a state law allowing the same thing. From the latter link:
She was sued and jailed for five days when a judge held her in contempt of court, but she was later released when other employees in the clerk’s office began issuing licenses.
The next link is to a story that is contemporaneous with her release.
Kim Davis did originally question whether licenses without her signature were legally valid. But I don't know of, nor can I find, any support for her opposing making such licenses valid. Her attorneys' suggestion at the time of her release was to make the licenses issued by the state instead of the County Clerks. Then Governor Steve Beshear (whose term has since ended) refused to call a special session of the legislature to do that.