A modern place which is a good compromise between luxury at high prices and a dump at bargain prices is the hotel Bauzá (calle Goya) which is usually about €150. It is an ultra modern place with smallish but comfortable rooms and a chic bar/restaurant.

I´m not sure how far east you need to be, but this place is east of the Castellana (which essentially divides the city in half) but is still very central and in a good neighborhood for your dinner/drinks.

You might try Teatriz (modern, somewhat upscale place in calle Hermosilla) for dinner and Juan Bravo 25 (very inventive tapas and convivial atmosphere) for lunch the next day. These are moderately priced places (which will seem downright cheap compared to London) where I have repeatedly had good experiences.