Is it permissible to repeat the supplication: "Peace and blessings be upon you and upon your family, my master, O Messenger of Allah, you are my means, take my hand, my strength has diminished, so reach me, O Messenger of Allah, pray to Allah for my healing"? And when does the Prophet become a means for his Ummah?

Supplication does not heal by itself. Rather, the Healer is Allah, glorified be His Majesty. Believing that supplication heals by itself is polytheism (shirk). A Muslim must believe that Allah is the Healer, and supplication is a means that may or may not be answered. As for supplicating to the Prophet, peace be upon him, after his death, whoever believes that he brings aid has committed shirk. If one intends to seek his supplication or intercession, then this is seeking intercession through him after his death, which is a matter of dispute among scholars.