A few ideas to kick around…
Puerta del Sol (Kilometre 0″ in Old Madrid)
Basilica de San Francisco el Grande
Plaza de la Cibeles
The Rastro (flea market)
Take in the footy? Or will there be matches on in November?
Nip up to Toledo for a day. I’ve seen that it makes a great day trip from Madrid, and I’ve had a thing for Toledo ever since 5th grade, when I saw the El Greco painting