I think you can find what you are looking for, but hardcore continues existing there. Lavapies is a funny neighbourhood, full of bares, hardcore pensiones and crazy young people. It is near to Barrio de las Letras, which is more classical (look for Huertas street in a map like lanetro.com) where people from Madrid and tourist use to go at night. You should not look for hotel or pension near Gran Vía Sreet (very dangerous for happy tourist at night). Look between Sol and Royal Palace, but avoid Atocha (same reason). Metro system and night buses work fine in Madrid so If you don´t find anything there, Barrio de Salamanca (Velázquez, Serrano streets) is a good choice, It is not far form Puerta de Alcalá an all that (15 min walking on averange). Avenida de America is the farest place you should look for hotel (20 minutes to downtown by bus). There are many hotels for 50€/double room-night with something called BancoHotel(banchotel.com). Enjoy!