After many exchanges, some of them in Spanish using Goggle translate (it is odd, but better than nothing for a new Spanish learner like me, we finally find a room in an apartment. The host is an eager man who speaks simple English and seems to use translate website to write English with us. We'll stay here for one month, when we shall be finding another cheaper place.
It only costs US$400 a month, with kitchen, shared bathroom, internet, parrilla. It is located at
435 Cochabamba (between Defense and Bolivar), Ciudad Autónoma de Buenos Aires, Capital Federal, Argentina
It is in San Telmo area. Wish us luck about the place. If you know other cheaper place in BsAs (especially near Palermo), please inform.