Don’t know London hotels at all (thank goodness) but surely obvious solution is to go to a site like booking.com...put in area of London you’re interested in, press the button and look at results.
Assuming you do use booking.com there’s a rating for each hotel from customer feedback, a negotiated discount rate, and ability to book large majority of hotels without inputting credit or debit card details.
Once at hotel itself..I doubt if any will refuse payment in cash, though I suppose in theory some may like to take credit card details as a long stop....but if it comes to that, easy enough to look in wallet, look perplexed and say “Darn I’ve forgotten to bring card.”