A sick person performing Salah as their health permits them
It is obligatory on a sick person to perform Salah as well as they can, standing, sitting, lying on their sides, or even lying on their backs, due to what the Prophet (peace be upon him) said to
`Imran ibn Husayn (may Allah be pleased with them both),
"Perform Salah standing, and if you can not, (perform it) sitting, and if you can not, (perform it) lying on your side."
and the words are quoted from
Al-Nasa'y) It is not permissible for any one to pray for someone else. However, if the lady you mentioned was mentally ill, she did not commit a sin and you are to be thanked and rewarded [by Allah] (Part No. 8; Page No. 85) for calming her and giving her the hope of a recovery.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.