What is required of a wife who used to pray the two rak'ahs before Fajr (Sunnah al-Fajr al-qabliyyah) and complete them, then intend them as an obligatory prayer after finishing them, and has done so an countless number of times?

Supererogatory (nafl) prayers do not substitute for obligatory (fard) prayers, and the intention must be concurrent with the Takbirat al-Ihram. Therefore, your wife's prayer does not count for the Fajr prayer, and she must repent and make up the prayers she missed until she is reasonably sure she has fulfilled her obligation. The husband is responsible for advising his wife, enjoining good, and forbidding evil. Abandoning prayer is a grave sin, and some scholars even permit a husband to strike his wife if she abandons prayer.