Free you handled the situation well. I work with adolescents and children and many have severe mental health issues and many self harm. A major reason for self harming is that young people want to feel something, anything else than the emotional pain they are going through, they can't cope with it so want to feel a different form of pain. It can feel soothing for young people seeing blood and they feel a release of their emotions and pain.
Have plenty of sterile wipes in the home so you or your dd can clean the wound, this will help prevent infection and if your not sure about a cut or whether it is deep take her to the dr or the emergency dept and they can assess it. One thing we do with our young people is draw up a safety plan, who they can talk to if they're upset, wanting to self harm or if they are worried they have cut too deeply. This usually works well for the young person? Sorry I know this sounds very practical.
There is nothing you can do to prevent this all you can do is what you are doing and that is reassure your daughter that you love her and you are there for her if she needs to talk. I would try not to tell her how it makes you feel, this may cause her to feel guilt and feel she is letting you down in some way.
This is not your fault in any way. its good she has a therapist hopefully she can talk to her about what's going on for her and why she is self harming but also the therapist should be able to discuss with her various coping strategies instead of self harming.
((((Hug)))) to you and your dd.