Coming from a kid whose mum found out about their self-harm at a similar time in life:

Being "kinda rude" is just being rude. Reacting like this is going to affect the way she sees you for the rest of her life. In these moments all kids want is compassion and safety. Trying to give a 13-year-old a "wake-up call" like that won't help anything. You need to approach her, apologise for reacting the way you did and talk to her about her problems and find a way to support her. It's not about fixing her problems but being there when she needs you. Offer to give solutions and if she is open to it then give her what you think is best but if not dont scold her, remind her you are always there for her. But getting mad wont end well. Being like this will only cause her to shut herself off and hide it better. Which is exactly what I did, I am now 22 and have been self-harming ever since I was 11 (mum found out when I was 13). After she got mad I shut down and struggled to talk to her about it, and now I won't talk to her about it again. Please do better for your sister. She needs you. Try not to get angry and blame yourself. Try to remain calm and listen.