We already have this, it's called housing benefit. A cap was recently introduced on the payments to deal with extreme cases (single mother with many children living in Zone 1 getting > national average income). It doesn't actually impose any extra responsibilities on landlords.

What we used to have was a national programme of building social housing. This had some high-profile social failures (qv US "housing projects") and the government embarked on a programme of selling it off at a great discount to the current occupiers while demolishing some of the worst tower blocks.

What we now have is a massive failure of planning policy. The symbol of this is the derelict for 30 years Battersea power station. It's a huge, useless, architectural treasure that's occupying a lot of land in Zone 1 because nobody can agree what to do with it.