3 Day LONDON TRIP

As london is one of the mot expensive cities your req is a tough one.
Consider youth hostels some have pvt rooms.
Gd luck or incr your budget then try travelodge uk, ibis or premier inns uk.

I'm fond of the EuroCheapo website, but you'll see that even they don't list anything that inexpensive for London: http://www.eurocheapo.com/london/. However, if a place listed on that site interests you, do e-mail the place directly to ask about their best rate for your dates.

If you really can't stretch your nightly budget, look at hostels via Hostelworld: http://www.hostelworld.com/ or Hostelbookers: http://www.hostelbookers.com/

Airbnb can be set to search for rooms to rent in people's private homes. I've done this several times and had great experiences.
I just did a quick search for 4 nights in June and found about 30 listings for private rooms for $72 and less. Here's a link to the listings that I found.Take a look!
I can't think of anything in London, other than this, that will be available for under $70 per night.

I remember last year I was searching for budget rooms in London, before my first visit. Anything under £100 or certainly around $100 was very small and basic. One listing caught my special attention as it listed the size of the room as 43 square feet. I realized that was about 6 x 7 feet, incredibly small! So, I upgraded my expectations. London is a very expensive city for lodging, and you get what you pay for. I ended up with a room that was more expensive, but several times larger and nicer!

Try the Hotel Meridiana near King's Cross.

70 dollars is 47 pounds and think about what (if any) sort of room you'd get for $47 in New York City.

70 pounds would be do-able. We found a great Air B&B in Regent's Park for $110/night (75 pounds) and that was some dude's second bedroom with non-attached bathroom.

Good luck.