What to do if my father passed away and his ex-wife, whom left 12 years ago, is still on the deed and she wants half the house?

Under NY law, when a married couple owns real property, they own it as "tenants by the entirety" & if they remain legally married, upon the death of one of the spouses, the surviving spouse becomes the owner of the deceased spouse's interest by operation of law.

However, upon the termination of the marriage by the issuance of a judicial divorce decree, ownership of real property by the ex-spouses is converted to what's known as a "tenancy in common". When property is owned by "tenants in common", the co-owners each have a one-half interest in the real property. Upon the death of one of the owners, the deceased owner's one-half interest becomes part of their estate, but the remaining one-half interest is still owned by the surviving co-owner. 

Based upon your description of the facts (you asked how to get her off the deed), it appears that your father's ex-wife retains a one-half ownership interest in the house, since she did not transfer her one-half interest to your father by separate deed before your father passed away. You can obtain a certified copy of the last recorded deed from the county's land records to confirm ownership. 

Your father's estate retains the other one-half ownership interest in the house. If you wish to resolve the issue of the ex-wife's ownership interest, you will need letters of administration (if your father passed away without a will), or letters testamentary (if he had a valid will & appointed an executor in the will). Once proper letters are issued by the Surrogate's Court (the letters must state that any restrictions on the power of the executor or administrator to sell real property are lifted by the court) the administrator or executor can attempt to negotiate a settlement with your father's ex-wife, which may result in either: a transfer of her one-half interest to the estate, a transfer of the estate's one-half interest to the ex-wife, or a mutually agreed sale of the house to a third party, where the estate & ex-wife have agreed what proportion of the net sale proceeds will be paid to each co-owner.

If a settlement cannot be reached between the estate & ex-wife, the administrator or executor can start a lawsuit in supreme court called an action for partition & sale. Under NY law, certain parties, including tenants in common, can maintain an action seeking a judicially ordered sale of the real property. The Court has the power to appoint a referee in the action to conduct an accounting, which determines the respective contributions for maintenance & upkeep of the house & how the sale proceeds should be distributed after the court ordered sale. When the referee completes the accounting & has reviewed each co-owner's admissible evidence & documentation (including satisfactory proof re: mortgage & tax payments & other expenses), the referee submits a referee's report to the judge, stating his or her findings & conclusions. The judge can confirm or reject the referee's report, or send the matter back to the referee with specific directions from the Court.      

You may wish to retain an attorney who practices in the areas of real property & estate law, particularly if the ownership of the house has to be transferred by written agreement, or if the estate or your father's ex-wife decides to commence legal action. This is a complex legal matter which requires proper evaluation & representation by an attorney based upon the specific facts & the applicable law.