Legal Question in Credit and Debt Law in Kentucky

Depends on what the Property Settlement Agreement/Decree/Court's Order says about it, and what jurisdiction the matter is in. Likely, the ex wife should be responsible for the payment, and could even be in contempt of court, depending on what was ordered.

However, when the orders are silent on the matter of refinancing, it is questionable whether the Court will find that the spouse keeping the house must refinance when the matter is brought before the Court in a post-divorce motion. One party may have executed a quit claim deed to relinquish their interest in the property to the spouse keeping the house, but then that spouse has failed to refinance the property, leaving the other's credit tied up in a mortgage for a house for which they no longer have interest. I've seen judges rule against parties whose former spouse kept the house and did not refinance, on motions asking the spouse to refinance. Hopefully your husband's divorce decree addresses the issue of refinancing. If not, she should still be responsible for the full amount of the back amounts, but she may not be forced to refinance.

It would be best to take the pleadings from the divorce to a family law attorney who practices in that jurisdiction and seek a consultation to go over your options. Many attorneys offer free consultations. Waiting any longer to address this may do further damage to your husband's credit.