Of course there's ways to do it, I was going under the conditions set forth in this thread. High rises are out, for example. And that's a tried and proven method of making more housing available on any given space. But apparently London doesn't want to do that.
So now you have an absurd situation where 5 million households want to live in an area with only 100,000 units of housing (I'm using a bit of hyperbole here to prove a point, I have no idea the actual numbers). There is no free market way to make that affordable, that's for sure.
So perhaps you wish the government to buy the land and build the housing in order to make it affordable. Which they could certainly do, but that does nothing to alleviate the shortage. You still have 5 million households chasing 100,000 units of housing, presumably there will be a lottery of some sort to decide who the few lucky ones are. End result is there's still no place for the vast majority of low-skilled workers to live in the places they're most need to work, and they're still resigned to a long commute from the outskirts housing they can afford.
By building high rise residential buildings you increase the supply of housing in the central area, which takes the pressure off housing prices. And while the low-skilled workers might still be unable to afford it they should at least be able to afford to live nearby. If high rise construction enables 100,000 more people to live near where they work that's 100,000 less commuters clogging the streets, trains, and buses every day.
Basically, any solution that doesn't include a long-term plan to encourage high rise construction is doomed to failure, whether it's public or private housing. Yes, it will drastically change the way London looks but it would make it a much better place to live. And the construction boom would last for decades. And no, it wouldn't have to look like Hong Kong!