We stayed in Cortes (in the old town) last Easter, very central, safe to walk around at night, loads of cafes, restaurants etc. Botanical gardens and the art gallery (where Goya paintings are) whose name evades me at the moment, about a 5 min walk away. Reitro, huge park with cafes pond with boats, etc bit like Hyde Park is a 10 min walk away.
I agree ditching the hire car, we walked everywhere.
There was a Radisson Blu Hotel around the corner from the apartment we stayed at, we did go to the cafe at the bottom of it for a coffee one afternoon. Looked fine.
We stayed in a fab 2 bedroom apartment in a beautiful old building, had stairs and a lift. It was a recommendation by a fellow mumsnetter.
www.spain-select.com/en_US/rent-apartments-madrid/moratin-i/photos
The second bedroom for the kids had a pullout bed under the single bed. We paid €135 per night.
They have a fab concierge service where they can book restaurants, tickets to events, anything really and its 24hrs service. Was useful when we got stuck there (late at night) for those 2 extra nights due to air traffic controller strike!
They can book a cab from airport (only 30 mins away if the new one, old one was 20 mins away) which we did but they charged us €70 when we arrived but on way back we found using local one about half the price e.g. €35 euros one way. Easy to hail down in the street.
Wear comfortable walking shoes.. Oh and elasticated trousers cause you will be eating (and drinking) loads it was so cheap, well it was last year.
Not sure if it's your thing but we did a Great walking tour with Ogo Tours Madrid, free but you tip the guide what you think it's worth. About 2.5 hrs. I found them on trip advisor, fab reviews and worth it, gave us lots of local tips to cafes and restaurants that locals go to.
Anyway hope this helps, very envious would love to go back again.👍