It appears to me that KY allows the State to request a change of venue.
When a criminal or penal action is pending in any Circuit Court, the judge thereof shall, upon the application of the defendant or of the state, order the trial to be held in some adjacent county to which there is no valid objection, if it appears that the defendant or the state cannot have a fair trial in the county where the prosecution is pending. If the judge is satisfied that a fair trial cannot be had in an adjacent county, he may order the trial to be had in the most convenient county in which a fair trial can be had.
My feeling is that it is going to be extremely hard to find an unbiased jury in Letcher County. The population of the whole county is only 20K, so it's a pretty small pool to start with. Then to find enough people who've have no biases and have expressed no opinions publicly about Stines guilt or innocence will be like looking for a needle in a haystack.
I think this is especially true after broadcasting the shooting video. IMO it was a huge mistake. Instead of tamping down speculation and gossip, it just turned things up.