I can not remember that it has been mentioned that it is one hostel/hotel in particular that "all" pilgrims stay at in Madrid on this forum. If there is one good one that someone can recommend, please post here.
If you click the link below, you will see a list of hotels close to Plaza del Sol in Madrid. Have a look at the different ones, and pay especial attention to the reviews that people that have stayed there have written. You will get a good idea of the place, just by reading those reviews. You will find the link to the hotel-review towards the right on the screen, just above the bar that indicates the user-review-score for that hotel.
Keep in mind that Plaza del Sol is downtown-downtown, the center of Madrid and you also pay for this.
Alternatively you could look for a place around the Atocha Train station (1km away, a short walk away) and you will find that prices should be lower. By the way, Atocha is close to El Prado... and from Atocha it makes for a great walk up past El Prado and up to Cibeles and then Calle Alcala to Plaza del Sol....
Sorry that I can't give you any specific recommendations.... but I hope it helped somewhat.
Last time I was in Madrid we did not book a hotel or hostel.
We simply walked down the main street in Madrid and there are signs above the sidewalk that say they have rooms for rent or "camas" - What that means is that the person has a room in their own apartment or house for rent. Usually this is a very clean room at a VERY good rate! We never paid more than $35 for two people and stayed in some lovely places.
It's perfectly fine to ask to see the room and then decide.
It's also ok to negotiate a price as long as you are fair and not obscenely low in your offer.
I'm not sure about high season, but we were there during Holy Week and had no problem at all finding lodging.