Prayer works according to the good sovereign plan and will of God. Prayer does much more than perform public miracles, it performs many unseen miracles. It demonstrates our limits and demonstrates our desire to rely on an all powerful God to affect lives and things outside of our control. It invites us into the presence of a Holy God on the basis of Christ alone, knowing that without Jesus, being in the presence of God in our sinfulness will only condemn us. It gives us an opportunity to wait and see the goodness of the Lord as He answers us in ways we never would have imagined.
There is a need for doctors, there are many unseen miracles going on in the operation room. With each tragedy, pain and success, the hearts and lives of doctors, patients and nurses are all being affected. God has also given us science and this world for us to know much of it and how it works in order for us to worship Him in all his beauty. There is so much we can learn from the human body and how it works and how to heal it. God and miracles are not a substitute to science. We never stop learning about this world and the precise workings that God has put in place. The statement you made can be applied to anything in this world. If there's prayer, why is there a church, why do we have missionaries, why do we have engineers, why can't we just pray and have everything instantly happen the way we want it. The answer to that is, prayer is a means of grace that God uses to accomplish His will on the earth. It can be answered in many ways, but it is never a substitute for action. We pray and act. We pray and live. We pray and play. We pray and eat. We pray and glorify God.
God wants us to persist in prayer as well. Never stop praying for your mom and loving her and being a Christ-like daughter to her. Your consistency, your tears, your prayers will all be heard by God and at the right time He will bring you healing and comfort and will wipe away your tears.