The slight problem Ant is something little thought about in the UK
Liability insurance (Wikipedia)
You need to have that to even sublet someone's room. Why? Well if you flood the bath or go on fire or the likes then it has to be covered. You need to get registered and the likes to get it unless you get very very lucky.
I know a guy who didn't have it and he fell asleep with the bath running flooding the Leather Goods shop that was below on a Sunday. Today he is 50,000 minus as a result.
I also know a guy who knew a guy who knew a guy that went to the moon... Anyway, don't scare the guy.
You should register as living somewhere, but that's step C. You're at step A. I've been currently looking for a flat as well. Same kind of conditions (dates, etc). Now is the easiest time of the year to find a flat, because people sublet their rooms for the summer break, where many people take off and go somewhere.
Remember you're not trying to impress somebody's grandma with your crap german. Basically everyone speaks english in Berlin. Just write in English, say you're learning german, and hope you can speak entirely in German unless there are words you don't know or if their patience is running thin at that moment. It takes a lot of patience to carry on a conversation with someone who barely speaks german.
There really are tons of places available at wg-gesucht.de or studenten-wg.de . I suggest you keep playing around with the search criteria until something pops up. If you want something cheap in Prenzlauer Berg, that might have been the fatal flaw - doesn't happen too often. I've seen people trying to rent closets for €300 a month.
Good luck!