Suggest telling her dad when she is not around----and, although his denial is difficult, I would provide him with some information (written? other...) on what the behavior "means". She needs to know that his yelling/being angry is his defense, his problem, and no reflection on her; he does need to know for practical reasons, "what to do if..."
Her anger at being "outed" is predictable...and, as difficult as it is, it is a place to start...
she also needs information on her own behaviors/feelings---
I don't know if I would search her room unless I felt she was suicidal...and then I would keep it low key...she needs to learn how to manage her emotions, you can't be running after her keeping her safe 24/7---there is always a risk involved, but I think it is riskier to take responsibility for her safety----she says she doesn't plan on doing it again---I would let her know I will take her word for that but, should it happen, or the urge becomes strong, that you know the feeling and are there to talk with/request help from.
If talking right now is inflammatory, a brief, kind, empathetic (not sympathetic or distraught) note might be best.
I would talk to yr. t. again, find out how familiar they are/comfortable they are in dealing with the kind of behavior your daughter is displaying. It is good that the school knows (hopefully, they will someone on board who can offer a thoughtful, appropriate support system)----
All the best to you....hang in there...