Long Term Care Insurance Integrity Act of 2013 - Requires an insurance issuer that offers a long-term care insurance plan to implement claims dispute resolution procedures that: (1) are designed to expeditiously resolve disputes; (2) provide for alternative means of dispute resolution involving independent third-party review by entities that are mutually acceptable to the issuer and the enrollee, with the reviewer's decision being binding on the issuer; and (3) ensure that an enrollee is eligible to obtain claims review only to the extent and in the manner provided for in the applicable insurance contract. Allows an enrollee to appeal the decision of an independent reviewer to an appropriate state court as provided for under state law.