Is it obligatory to secretly give a mentally ill patient medication in their juice, knowing that this medication causes them to sleep through prayers?

There is no harm in putting medicine in your brother's drink, even if it causes him to sleep. You must wake him for prayer when its time arrives, according to the Hadith of Abu Qatadah. If it is possible to administer the medicine in a way that does not conflict with prayer times, that is preferable. If he sleeps due to overwhelming sleepiness or the medicine, and does not wake up despite attempts to awaken him, there is no sin upon you or upon him; for Allah (Exalted be He) says: "Allah does not burden a soul beyond its capacity," and for the saying of the Prophet (peace be upon him): "Indeed, there is no negligence in sleep; rather, negligence is in wakefulness. So, if one of you forgets a prayer or sleeps through it, let him pray it when he remembers it." It is permissible to sleep before and after the prayer time if it is highly probable that one will wake up before its time expires. However, it is forbidden to sleep after the prayer time has begun if it is highly probable that one will miss the prayer and no means have been taken to wake up.