Since they share a faction Ryouma will be interested in getting to know Emily and also because her being a doctor is a very useful thing to be! Knowing who has good ideas and who has what knowledge is a good thing to keep track of and of course, he'd offer to help her out with anything she needs.

Also: he is going to be wanting to know all the things about stuff from slumscape if he learns about her daitengu because he's very invested in solving Izō's Karasu problem and can't ask Izō anything directly. (':