This question already has an answer here:
The main big hotel search websites I know: http://www.booking.com/ http://www.hotels.com/ http://www.laterooms.com/ list many "hotels" which also sell single beds in dorm rooms. Try searching for London for 1 person, and sort by price, you'll see that the cheapest offers are mostly single beds in shared dorm rooms. Its impossible to find the cheapest offer for the whole private room without examining each offer manually.
You could filter out "hostel" style properties, but that misses out on many facilities branding as "hostels" but having cheap private rooms, so its not an ideal solution. And on Booking.com even that won't work, as it still leaves many which brand themselves there as a "hotel", but have shared dorm rooms, and prices are shown for a bed in such a dorm room, and not a whole private room.
Is there really not a website which has cheap hotels (regardless of if they call themselves a "hotel" or a "hostel", as often this division is blurry and arbitrary, and doesn't specify what type of rooms the facility has), and sorts the the price of the cheapest whole private room?
The website should list most hotels, and not only hostels - websites like hostels.com hostelz.com hostelsclub.com hostelbookers.com hostelworld.com gomio.com filter by private rooms, but they don't show most real hotels (many of which have sometimes quite cheap offers), only hostels. These hostel websites list only max. 100-300 properties in London, while Booking.com has 2359 and Hotels.com has 1614, so a website listing hotels should have at least a thousand in London, as opposed to only 300 hostels.