Every question I've ever answered on this site, the answer has come to me offhand without much thought. Maybe some answers were pretty good and some were merely passable, but either way I knew what my answer was immediately without any further consideration.
This is a little different. It's a very, very difficult question. It's a question I've wrestled with before but never brought up here.
My initial reaction is Don't because, as I said in my comment above, nobody wants to read that kind of language. It's the same reason a lot of readers are very turned off from A Clockwork Orange. It's very inaccessible. If people can't understand what they're reading, they don't bother. Would you pick up a book written in Swahili? It's no different.
Really, the only difference is that this dialogue is only a portion of your book; the portion where this one character speaks. So it's a little better than in some other circumstances. But it's still a problem. Even if it comes intermittently in small pieces, readers still get irritated by incomprehensible passages of text. It really bothers me when I can't understand something, I'll tell you that. Kind of like daddy's little princess flying economy for the first time. WHERE'S ALL MY LEG ROOM???
Anyway, it's hard to come up with a good answer to your question. Even if you hire an expert on medieval language to maximize authenticity, we still hit upon that same problem I just outlined above: the reader won't understand it and certainly won't appreciate your misguided efforts.
The way I see it, this leaves you with a few options. You have to be clever. Incorporate translating technology, or have the character speak through a hired interpreter (and only occasionally do you actually write what the character himself said, in the cases where it's similar enough to modern language that the reader will understand and perhaps find some humor in the distorted familiarity in the words), or have the character speak very, very little out of shyness or shame. Or have them practice/be tutored extensively "offscreen," so that each time we see them they can say a little more, and it's realistic and acceptable that they've made this progress.
That's the best I've got. What do you think?