Adding my suggestions to Nikita's very helpful pointers....
It's not going to be easy, but then you already know that. But, yes, it's just about possible for someone like you who is clearly frugal in their lifestyle.
Bar work may fit in with your part-time job. Not great pay but good English-speaking staff are always in demand.
Things which may help eke out your budget:
Use local libraries for internet access (and to keep warm on days you're not working!) This will save you money on heating.
Cut your transport budget immensely by getting hold of a bicycle. Will a friend or relative lend you one? If not, register with your local freecycle group and look out for a bicycle and a lock for it; also a helmet. I would love to say 'buy a new helmet', but if you can't afford one, 2nd-hand is better than nothing.
If you have anything to offer first on Freecycle (I've seen all sorts offered (magazines, even empty jam jars!) this will make you more credible with the other group members and you can then post a "wanted" notice for a bicycle. Also join the Freecycle cafe group if there's one in your area: people there are always swapping skills, time and other free stuff. There's usually a good community you can tap into: help and be helped!
You can get free help to repair and maintain bikes in lots of places in London. The people at Hackney Bike Workshop are great and I'm sure will recommend if there's another bike hub nearer for you. http://hackneybikeworkshop.com/
Food. Find your nearest Aldi, Lidl, Asian supermarket and/or fruit & veg market. All these are cheaper than any of the big supermarkets for a lot of stuff. Get to know the basics ranges of Sainsburys etc: quality is absolutely fine on most things (flour, cream cheese, canned tuna, eggs are some of my staples). If you're in the City on any Sat afternoon, check the various big-name supermarkets for sandwiches & ready-meals at crazy-reduced prices (20p for BLT etc).
Porridge is my great economy! A 1kg bag of oats from the supermarket costs £1.10 (Sainsbury's price; may be cheaper elsewhere)... I measure 25g for a portion, add water not milk plus a good shake of cinnamon (buy cheap from Asian shops) and a teaspoon of cheap marmalade, bring to the boil and leave it to stand for at least 15 mins to fluff up before bringing back to the boil and serving. Keeps body and soul together more healthily and cheaply than any other breakfast I know, and the oats keep for months.
It's not all gloom and scrimping. Enjoy all the free stuff London has to offer! Museums are almost all free and often so are talks and other events there. There are free lunchtime concerts in many churches (with a collection at the end, but an appreciative audience is welcome even without contributing) and there's often free music in the lobbies of the Festival Hall/Southbank complex. Big events like the New Year's fireworks, Thames Festival, Lord Mayor's show etc are all free. Even paying gigs can be really cheap: my best bargains this year have been some brilliant sessions at the Water Rats (Grays Inn Road) with tickets at £8.
Anyway, good luck. If you have other questions, you know where we are.
=)