Is it permissible for a worshipper to recite the details of his daily affairs and personal supplications during prostration?

If what you do in prayer is merely talk and a narration of your life details, then this is impermissible and invalidates the prayer; because prayer is not suitable for any human speech, but rather it is glorification (tasbih), magnification (takbir), and recitation of the Qur'an.

However, if you mean supplication (du'a) and mentioning your state during prostration, then there is no harm in that, as prostration is a place for supplication, and it is permissible for a Muslim to supplicate for their worldly and otherworldly needs, as long as it does not reach the extent of transgression in supplication.