Please help me with appropriate prayers for my mother who doesn't pray or fast and also drinks.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The Muslim should supplicate Allah for himself first, then for his family and for all the Muslims with the following supplication.

Allah Says (what means): {My Lord, make me an establisher of prayer, and [many] from my descendants. Our Lord, and accept my supplication.} [Quran 14:40]

Moreover, the Prophet instructed Mu’aath Ibn Jabal, may Allah be pleased with him, never to abandon saying the following supplication after the prayer: اللهم أعني علي ذكرك وشكرك وحسن عبادتك. (O Allah, help me remember You, to thank You, and to worship You in the best of manners.)

Know, dear sister, that the door of supplication is open. So, you can supplicate Allah with any words you want provided you do not ask to get something forbidden or to sever kinship ties. The Prophet said: “The supplication of a person is answered as long as he/she does not supplicate for a sin, or for the cutting of relations with kinship, or does not become impatient. It was said: “O Prophet of Allah! What does “become impatient imply? He replied: “It is when he says: 'I supplicated and I supplicated but my supplication was not answered. Then he becomes frustrated (in such circumstances) and abandons supplication.” [Muslim]

So, trust Allah and be sure that since Allah has guided us to supplicate Him, He will answer your supplication.

Let your mother listen to cassettes that talk about the importance of prayer. In fact, the prayer is the relation between the slave and his Lord. If a person abandons the prayer, he abandons Islam. The prayer is the covenant between Allah and His servants; so anyone who gives it up becomes a Kafir. Try to have her read our previous Fataawa about the obligation of fasting and the prohibition of alcohol.

For more benefit, please refer to Fataawa 82187 and 81461.

Allah knows best.