Hi this more for advice for what I did wrong and where I can go.

So couple of days ago a girl from my Uni followed me on Twitter, slid in my DMs and we had a casual chat about Uni stuff ( where she lives on Campus general interests etc) and turns out she lives a floor above me. I said she seems cool and we had good chemistry then said weshould hang out. Yesterday we met at the library by coincidence and it was the first time we spoke in person at like 2am. Small talk and there was a lot of awkward silences and she then let to go do her work (she was on another floor of the library). I think she's pretty cute and we have similar interests, she has already expressed she's bored with her roommates. Where do I go from here. I feel like I bombed the first encounter it was at 2am, I was tired and didn't really have much to say and although didn't say anything stupid it just felt dry. Also term ends this week in which we'll both be going back home from Uni. Do I bother pursuing or do I wait for next term? What could I have done better?