How should one deal with an elderly, sick, bedridden grandmother who sleeps most of the time, cannot focus on prayer, and misses performing some obligatory prayers due to her deep sleep?

If the grandmother has lost her mind due to old age, she is not obligated to perform acts of worship such as prayer and fasting, because reason is the basis of obligation, and "the pen has been lifted from the madman until he regains his sanity." However, if she is sometimes lucid and sometimes not, then she is only required to perform the religious duties when she is lucid. It is permissible to remind and awaken her for prayer when its time enters.