Now here's the scenario.

At the college martial arts group, there's this new girl who's just joined - a couple of weeks later than me.

I've received what I think are IoIs from her.

She meets my eyes, laughs along with me, and actively approaches me to talk during the sessions. I've been able to slip in a little subtle kino ( I think). I've been able to have some time alone with her as we walk back to our homes, which are in the same general direction.

But here's the problem. The girl is question does not have a very good command of English. I do not speak her native language, and she does not speak mine, which is bad, because my English is better than her's and much of my dialogue is wasted.

How would I continue from here?

Actions are more potent than words anyway. You said you THINK you've been able to slip in a little kino. That means you haven't. Make sure you are doing this since, while it's incredibly important to do for any woman, it's even more important in this scenario. I'm sure she speaks decent enough English to understand something like "Let's go get dinner", right? And while doing so, I'm sure she understands basic questions like "Where are you from?", etc. So get her to go out with you, get to know her, and you can bet she'll appreciate that you took the time to look past the language barrier.