Most Helpful Girl
If you'd like to be then just have the conversation then send her flowers after making it "official". If you know her B-day I'd take that as a good sign and she is probably just waiting on you or assumes if you guys are spending several nights together.
Most Helpful Guy
Well without knowing you or her, i know a lot of girls flowers is a big deal and with just general advice would go to saying "don't send flowers."
You guys seem new and because you're gut is telling you "this might be a bad idea" i'd listen to it and just do something simpler. It's less stress for both parties.
simply dinner + card or a nice gift would suffice.