Timing will have an impact. Is this a summer trip? Are you willing to risk having no air conditioning? That's not likely to be available in a hostel; even many budget hotels don't have it. You might well not need a/c, but on the other hand, there might be a heat wave.
Saturday night is often by far the costliest night of the week, so if you can avoid spending that night in London, you may save money.
Staying outside the center of town--which I define as roughly the area circumscribed by the Circle Line on the Underground--can be quite a bit cheaper. That guideline will probably apply to hostels as well as to hotels and apartments, but now you're farther away from quite a few sights you may well want to see, and the Underground fares for two people will add up.