All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, may Allaah reward you for being keen on being steadfast on the religion of Allaah. One of the signs of being steadfast on religion is to be kind and dutiful to your parents even if they mistreat you or neglect your rights. Being kind and dutiful to them may earn you a good status with them and therefore they would accept your advice about your sister and anything else. For more benefit, please refer to Fataawa 91146 and 87019.
If what you said about your sister is true, then she is wrong. However, you should be patient and kind with her as Allaah may reform her, thanks to you. If she becomes righteous, then all praise be to Allaah, otherwise you may desert her if you expect that this will be beneficial. For more benefit, please refer to Fataawa 86090 and 85457.
Furthermore, you should advise your father in a good manner and remind him that he is her guardian, and that he is responsible about the behaviour of your sister and that he should be firm with her. If you do so, you would have done your obligation.
You should not hit your sister because it seems that there is no benefit in doing so. If hitting your sister is what irritates your mother, then you are not permitted to hit her. Slapping the face is forbidden in any case as confirmed in a Prophetic narration.
Allaah Knows best.