I will also be going to Birkbeck university in October, I am 22 years old and I have lived in London all my life. Drop me a private message if your still looking for something, perhaps we could look together, we would probably need 1 or 2 more people as most places are 3-4 rooms.
Oh and P.S not sure about my budget, but the cheaper the better, you only get so much SF.