There are several laundromats not far from Gran Vía area. You will find them at the following streets:
Pelayo, Barco, San Joaquín, Hortaleza (84), Válgame Dios...
However, the cheapest laundramat I know is located at Cervantes, 1, near Plaza de Santa Ana.
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The best tips from your favourite hostal in Madrid.
Hostal Chelo at
http://www.chelo.com