Where does taqwa (piety) actually reside?

In the heart. The Prophet ﷺ pointed to his chest and said three times, “Piety is here.” It is not our bodies or our forms that concern Allah, but our hearts and our deeds.

In a lengthy narration the Prophet ﷺ said, “For sure, it is not your bodies or forms which concern Allah, but your hearts and your deeds. Piety (Taqwa) is here! Piety is here! Piety is here!” And he pointed to his chest. [Bukhari, Muslim, Abu Dawud, Tirmidhi]

Ali may Allah be pleased with him narrated: “There is no real benefit in recitation of the Qur'an in which there is no reflection, nor in any act of worship in which there is no cognition.” [Ad-Daylami, al-Firdaws 1/135]

Allah created man to recognise his Lord, and this cognizance is not bound by time, age or space. It is a connection for all times, not restricted to the mosque or prayers. The inner state of the heart has to be practised and developed, because it is not taught by the books.