What is the ruling on the Prayer of Need, described as twelve rak'ahs, with the recitation of Al-Fatiha, Ayatul-Kursi, and Al-Samad in each rak'ah, and uttering the specific supplication in prostration after completing it?

The hadith with the wording mentioned in the question was not narrated as a marfu' (traced back to the Prophet, peace be upon him). Al-Ghazali cited it as a mawquf (وقف) narration from Wahib ibn al-Ward (وقف: traced back to a Companion or a Successor, not the Prophet). A marfu' hadith regarding Salat al-Hajah (Prayer of Need) with twelve rak'ahs and the recitation of Ayat al-Kursi and Surah al-Ikhlas was narrated, but scholars of hadith deemed it weak, and Al-Albani judged it to be fabricated (mawdu'). It is not appropriate to perform acts of worship in this specific manner due to the absence of an authentic hadith supporting it. The best approach is to limit oneself to the Salat al-Hajah that has been authentically reported in the Sunnah, for the best guidance is adherence to the Sunnah of the Prophet (peace be upon him).