Do you have any recommendations for housing?

Will be happy to give rec! When I did the ttmadrid TEFL course, I just used the company they recommended which was Sal y Ven housing in Madrid. Was good housing but pretty expensive so ended up moving after a few months. You can find their company on facebook:
https://www.facebook.com/salyven.alojamientosmadrid/
Another great website is idealista.com bc you can narrow down price range and which zone. I recommend La latina, Malasaña or Chueca neighborhoods for young people or students or anyone who doesn't mind a little extra noise.

I would recommend Sal y Ven as recommended by TtMadrid for the time being while you're doing the course. Once you've completed the course, you can look for another place depending on your work location. That time I shared the apartment with two flatmates (who were also doing the course as well). The apartment was fully furnished and equipped (TV, internet access, washing machine and etc). 5-minute walk to the metro. So the rent was considered all right for me. If you have enough time, you can find a place suitable to your preferences at idealista.com. I would say internet access is very important and location-wise preferably close to the metro. Hope this helps.