Can I give my cat something to make her sleepy without making her sick or hurt?
The first question here is, why do you need to make your cat sleepy?
We definitely here miss some more information to help you.
Anyhow, without seeing and examining your cat, it's not possible to prescribe you any medicine via online. Visit your vet who will perform a checkup of your cat and according to the diagnose, give her an appropriate treatment.
And remember: Don't self-medicate your pets. Keep in mind that not knowing of medications and their compounds, as well as the effect they have on our pets can produce serious harms.
All the best.