Having completed a four night stay on points earlier this month, I would like to add some info on breakfast options. If you are paying for your stay, breakfast is included. But if you are staying on points and are not a Globalist, breakfast is not included in your rate. I was in that situation as an Explorist.

The in-house restaurant Omar charges 25 for continental breakfast and 40 for full breakfast (the menu is posted in an earlier post above). At check-in we were told that if we wanted to have breakfast on-site, they would charge us 35 euros rather than 40; this discount was available only for the full breakfast, however, not for the continental. At the time we did not know, nor were we told at check-in, that Omar had a 25-euro continental option, which we might have taken. We declined the offer and ended up looking elsewhere.
It's possible that if you take the 35 euro option, you'll be charged through the hotel rather than by Omar, in which case maybe you would earn WoH points. I did not think to ask that question.

As others have mentioned, there are lots of breakfast places within a block or two of the hotel in any direction you go. However none of those places or their menus looked appealing to us. Our favorite place was La Rollerie, a small chain of excellent bakeries which serves a very good breakfast. There are two La Rolleries within a 5 minute walk of the Thompson, one near Plaza Mayor and the other on Calle de Atocha. Their pastries are first-class and freshly baked. They also have an a la carte menu with fresh squeezed juice, cappuccino, eggs to order, etc. and you'll be hard pressed to spend more than 10-12 euro per person. Worth considering if you are in the same situation we were.

The hotel is great and is a steal at 15,000 points per night.