-do not spend too much time in London, especially if you are on a budget
It's expensive as hell
I will just emphasise that it really doesn't have to be. If you can get decent accommodation, then you can have a really cheap and great time in London.
Sure, if you're a tourist and take a taxi everywhere, eat in the Angus Steakhouse at Leicester Square, and want to drink a lot of alcohol in upscale Covent Garden bars - then it is expensive.
However, if you get the tube, travel is fairly modestly priced. Last night I went to a great Japanese restaurant in china town and had a huge chicken katsu udon for £6. You can get a big salt beef bagel from brick lane for a couple of quid. McDonalds is cheap if you're on the go. Get a meal deal from Sainsburys. Go for a drink in Wetherspoons.
I moved to London from a City in the north, and aside from my rent - day to day expenses aren't much more. Nandos, McDonalds, Sainsburys all cost about the same - and I have a never ending stream of great, cheap restaurants and food places of every type of cuisine to try.
Most of the good tourist stuff is free.