1. Remove density and height restrictions (e.g. "Berliner Traufhoehe").
2. Remove unnecessarily strict building requirements (e.g. "Daemmungswahn").
3. Remove legislation that is too tenant-friendly (e.g. "Mietbremse").
All three serve as breaks to increasing housing supply, the first is a legal impediment, the other two reduce investments.
Doesn't really help getting more affordable flats on the market.
I'm not sure how familiar with the London property market. London is currently seeing a large amount of real estate being built, mostly studio, 1, 2 and 3 bedroom. Hardly luxurious. Prices are commensurate with London income levels. Otherwise people wouldn't live in London.
You also can't arbitrarily increase density of a city as you need infrastructure to go along with all those people. Not to mention that Berlin has many old buildings that are under protection, so you can't just demolish buildings and replace them with skyscrapers.