If you give them too much medication or the wrong type, they can get really weak and tired and they'll naturally go to sleep by themselves.
The problem isn't their energy though, "Extremely Weak" means their health is really low. You can help this with items such as the fruits for praising them, vitamins and hand gel. If you can afford it, doctor consultations are good as well to see if they have a serious or grave infection.
They get their rest by themselves when you close the game at night. As long as you don't open the game up again while it's still late at night, they'll be well-rested.
Good luck!
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