Are you familiar with clubs and bars located in East London?
When you're on a budget and you want a good time in London, you need to think in terms of north and south of the river. Go south and you're heading for the divide that is middle class/upper class dominated so consequently, those venues down south will charge you accordingly. Central London is and will always remain expensive and North London you'll notice will leave you with enough change to feed yourself after a night out but the problem there is the cultural diversity is such that people have formed communities and its harder to find appealing clubs there. Hence I recommend you look at East London as it is student dominated, relatively less expensive catering to people who are renting and the night life can be fun compared to other places.