Iberostar Las Letras Gran Via, we stayed here and it is the best one in this area with a pool(plus on gran via with such affordable price). we loved it.
Apartamentos Leganitos, we stayed here too and this ones also very good in such low price range compared to the areas they are in. Gran via is like the heart of Madrid.
Mercado de San Miguel, Madrid's oldest and beautiful covered market. This was one of our favorite experience here.
Entertainment & Night Life:
Teatro Flamenco Madrid, Qualtiy Flamenco dance song and guitar and the prices are also down as compared.
Sala el Sol, its a salsa club, they also give salsa lessons before.
Tips: 1. Do not take day passes for transport within the city as many places are at a walking distance if you're living at gran via. we got 3 day pass and it was a waste. As there are no metros or buses to many of the places.
2. Omio.com best website for all your internal travel bookings in Europe.
3. For internal travels in the city you can purchase day passes at any kiosk at the railway station.
4. Google maps is very handy here you just have to put your destination on google maps and it will tell you which train on bus or tram you have to take to reach there.