The
Hostal Victoria just off the Puerta del Sol was great in March when we stayed there. There is an elevator and it's very safe(you have to get buzzed in through two doors to get to your rooms).
It was really easy on the budget too. I think we paid about 60 Euros for a nice, very clean room with 3 beds. The metro is just a few steps away and breakfast is less than a block away. The lady who runs is probably also a senior (although she has a husband and a couple of young people helping her out).