Alright so this actually doesn't look too bad. Tbh I think you trying to "strike while the iron is hot" wasn't a bad move. I think you probably could have gone about it in a better way by asking her what her schedule is instead of continue to suggesting times hoping that one of them would work (so something like "Sure, what's your schedule like?") - it's unfortunate that she replied the way she did but here's where the biggest fuckup came and I think you know what I'm going to say.
I think you took the law student thing way too seriously. She's just mentioning that she's busy, not that she's more busy than you, or that she's somehow getting less sleep than you and therefore can't meet. Your response was unnecessary and came off very bitter (reactive). I'm glad you're recognizing that but that's pretty much the only major fuckup I see here. I think you would have been fine if you had just said what you suggested or even just liked her message.
As for reengaging, I would wait at least 2-3 days, maybe a little longer and probably reengage with something like "How are you feeling?"
