Hi
I voted YANBU. People are getting understandably het up over what they perceive as a betrayal. I suppose reading a diary is a betrayal but it’s too late, it’s done, and now you’ve got to live with the consequences. You read it because you’re seriously worried, not because you wanted a laugh.
Guilt is a useless emotion to both of you in this situation. You know telling her isn’t the move, that’d be for you, not for her, and she needs your help, not vice versa. If you tell her you’ve been peeping inside her head she’ll likely retreat further within.
Sometimes we need to just live with the consequences of our actions alone. Use the important insight into her psyche that you learned through a morally dubious means to help her. Your fears are confirmed so start making plans for how you do that. Be subtle, don’t go in too obvious. You’re her sister, it’s not like you’re strangers, you have empathy, so use it. You were already concerned about her. Prove to her through action, not words, that she’s not an embarrassment and show her she’s important to you. You seem smart, it won’t be hard to think of ways to do it.
Also, worth bearing in mind that sometimes when people are writing when it’s never intended to be read by third parties that they can become very upset and overstate. As in, nobody loves me, I wish I was dead, that sort of stuff. They don’t necessarily mean it from moment to moment, throughout the day. Diaries are often a means of catharsis, they are not sworn affidavits, so don’t be unduly panicked, just chill and think about how to help her.