This is an unfortunate situation but I don't think you "fucked up", you simply didn't notice the screen recording was on.
Honestly, why wouldn't you talk to your therapist about your step-daughter? That's exactly what therapists are for and while I get she must have been curious, she invaded your privacy by watching what was recorded. Unless it's the first thing you said to your therapist, I don't see how she stumbled across that part accidentally. You said it yourself, it's your personal laptop, she should have stopped watching as soon as she realised this was a personal, confidential conversation between you and your therapist.

The damage is done however and she's upset, understandably so. It can't have been an easy thing to hear, even if her own actions have been the cause for it. Considering what you've said about her, I don't think sitting her down and having a talk will be very productive. If she would agree to it at all, it would end in another emotional meltdown.
Maybe you could write her a letter in which you could make it clear that as the daughter of her mother whom you love she is wanted and cared for. And that while things haven't been easy between you two that doesn't mean you want her to be unhappy or leave. Maybe offer to have an adult conversation at the end of it. No blaming, no apologizing, simply making sure she knows she has a place she belongs.

Also, maybe it's time to stand up for yourself a little more next time she treats you with disrespect. She is 18, it's time to stop being a petty brat even if she's hurting. It doesn't make hurting others okay. Maybe she'd benefit from therapy as well.

Just my 2 cents. I wish you all the best and that you'll both find a way to move on from this.