Was the deceased mother, while suffering from a mental illness that sometimes deprived her of awareness and perception, obligated to perform acts of worship, and is anything incumbent upon her children towards her after her death, and is her illness considered a weakness in faith?

Prayer and fasting are obligatory for the sane and capable. The non-sane is not held accountable. If the mother was sane, you should gently encourage her to pray and fast. After her passing, you must supplicate for mercy for her, seek forgiveness for her, and give charity on her behalf. The sick person is rewarded for their illness, and Allah expiates their sins through it. There is no evidence that psychological illness is a sign of weak faith.