Welcome, I would cheap and London do not go together in the same sentence. That being said, I think omisan and jw1313 have stayed at the generator if I remember right. There are thousands of hostels all over Western Europe, some better than others. Ask on here if anyone has stayed there or some places have reviews, pictures, etc. And if you're not happy with it, you can always leave after the first night. Usually they allow that and refund your money if you paid for subsequent nights. Or make a reservation for only one night to make sure you like it.
Flight to Tokyo: 92,000 yen
Metro to Ginza shopping district: 160 yen
Watching middle-aged Japanese woman dance to U2: priceless