Kentucky requires a special election for vacant seats. By the way, just to note, after January the Kentucky Supreme Court will have a makeup of 5 Democrats and 2 Republicans. Chief Justice is Republican though. All the more reason for the Senate to skip the whole thing and declare a vacancy. If the race was a close win for the Republican in a general election, then the Republican should take it handily in a special.
If it's a special election, she clearly would not be eligible. Part of the Republican argument is that Woodward should have brought the suit earlier, either before the primaries were over or before absentee voting began. Woodward chose to bring the suit the day before the election.
If a special election was held, they would immediately contest Stephenson's residency and prevail since she clearly did not meet the law's requirements.