I offer a room a person in a two-room apartment located at 1 min. from metro Nueva Numancia, in Madrid city centre. My apartment is not very big but it is cozy. The flat has a living room with a sofa, dining table, TV and air conditioning. The kitchen is equipped with all the appliances you need and the washing machine is in the bathroom. The person who would stay at my apartment has to be willing to take care of my two cats. They do not need special care, just to feed them, provide water and make them feel accompanied (just when you're staying in). In fact one of the cats will hide because she is very shy. Also, I may be coming and going from time to time for a few days.
If you want to experience the vibe in Madrid, don't hesitate. You can ask me all your doubts and I'd be delighted to answer.