Hola, MaryLynn!
In Madrid you can obtain credencial at Iglesia de Santiago y San Juan at the Plaza de Santiago (very close to Cathedral and Royal Palace). From your hotel it's app.20min walk (1,6km) NW on Calle de Atocha. Google it and you'll see it's really easy. Don't know the opening hours though. When I was there last year it was closed, most possible because of festivities in the time of coronation of a new king.
But maybe it would be easier to go to Asociación de Amigos de los Caminos de Santiago de Madrid:
http://www.demadridalcamino.org/
They are opened Tue&Thu 19:00-21:00, Wed 11:00-12:30 & 19:00-21:00. The address is Calle de Carretas 14 and is in the same direction as already mentioned but only 1km from your hotel.
Ultreia!