People are moving into London at a huge rate. That to me suggests that the city is doing something right, and should keep doing what it's doing rather than try to radically shake things up.

I don't think rent controls or second home rules can possibly work. That leads to people working jobs that aren't actually economically sustainable - and the benefit often ends up going to the rich (e.g. cleaners living in subsidized housing can be paid lower wages). A land value tax would be a good idea to encourage productive use of land, but the only way to avoid introducing loopholes is to apply it consistently to all land.

My view is that agglomeration really is valuable, that the high cost of housing and office space in London reflects their high value. Industries that really benefit from being in London will locate themselves there and be more productive as a result. Industries that don't benefit will move out and make space for those who can make better use of it.