M&N Residence offer month by month, no contract with one month deposit. No fridge or TV, but I believe they maybe able to rent also monthly. Located about 100m from Samsen Rd on Soi 5.

I would check homestay.com You could easily negotiate a month for 9k or even less.

Around the Khaosan area, you have banglampoo Place that have rooms with aircond, tv and cable, security safe, bathroom and small balcony. Bht15, 000 per month, one month deposit, cheap 8bht unit electric rate. 5 minute walk to Khaosan Road.

You can get places around that area Banglamphu and Samsen just down the road from Khao San Road for 1,500 Bath per month!

Of course, if it's "the cheapest" you're looking for, as indicated in the subject heading, at one time a number of people were "accommodated" under the Phra Pinklao Bridge (including one or two falang that I knew of).

There is actually an Israeli-owned hostel called the Overstay underneath and just next to the Pinklao Bridge on Thanon Charoensanitwong!