Any UK lawyers around or anyone who understand about private tenants rights?

Hi everyone
Just asking around as me and my partner moved to this flat in SW London which is £1600 per month plus bills, once we moved on 1st of august , the kitchen already had a big water stain in the roof (no one lives above we are the top floor) and the agency has been trying to fix it.
The living room also has a weird crack on the wall (all these things were there when we saw the flat) and it was all freshly painted when I saw the flat.
Anyway, once it started to rain, the kitchen one started to drip, and now it’s huge and even the plaster came down, wet patches started to appear around the flat, and others started to drip too.

The agency and land lord say it’s the block management issue, which we called them and they say they are trying to do it maybe next summer! But also, they want to change it to “priority” and to get enough funds to fix the whole roof.
It supposedly has an old water tank on the roof which has small holes which leads to the kitchen and the other areas of the flat im not sure maybe it spreads.

So the agency maybe will reduce the rent but we called shelter and they say to break from the contract (it’s 1 year contract) would be a bit difficult, only if it becomes absolutely unliveable and we would have to get the environmental officer to help.
Well we just would like to get out of here really as it’s crazy expensive and not worth it.

Any advice would be highly appreciated.

Thanks !