I did an "intensive" 3 week (3 hours per day, 5 days per week) Spanish course in México, extending it by another week and was in a homestay for three of those weeks.
I wouldn't do it again.
Coming close to 50 I really, really struggled to get back into "school" mode. I couldn't concentrate for long bursts, I became very tired and one on one the whole experience was too intense - there were no other students to benchmark myself against and no break at all.
Also, I was in a very beautiful and tranquil part of the world - this proved to be a major distraction when I should have been studying.
Madrid could be like that.
Furthermore, I did not enjoy the homestay with a couple. She was hardly there, he talked way too fast, refused to slow down and was openly critical of my (lack of) progress. Pretty demotivating.
I left with a handful of phrases, a skeleton of structure but without any flesh to add and more aware than ever of how much I didn't know.
The school was great and flexible, the teacher was lovely. The issues were all my own.
I've since vastly improved my Spanish by adapting Duolingo (the fora are great but Duolingo Spanish is designed for the Americas) recording vocabulary to test myself while doing something else and using youtube, Netflix, Podcasts etc. Short, fun 15 minute bursts work best for me.
It's interesting that you will be doing this after your Camino? I would have thought a couple of weeks in advance would be more beneficial?
Don't forget August in Madrid will be very hot and as many locals as possible will be out of town. It may not be the best of times in terms of teacher quality, homestay options or mixing with the locals. Also, learning in the heat can be tough!
Good luck!