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Hello, my name is Jim and I am a qualified lawyer happy to help you today.
By law, landlords must ensure the property is in "good repair" under S.11 of the Landlord and Tenant Act 1985.
It would be breach of contract on their part and you would be within your rights to terminate the tenancy agreement. I note the council have strangely not said the flat is uninhabitable - you may want to make a complaint about the council as mould is a health and safety risk - it can lead to respiratory illnesses so I have to say I am surprised at their stance.
If the council cannot help you would either have the option of paying it yourself and withholding rent or to instruct a tradesman to come out and fix the problem and you then pursue the landlord for the money. This assumes you want to remain living there - if not then you would have to cite breach of contract on the landlord's part and move out - then sue the landlord for your rent (since the mould was discovered).
You can sue the landlord in the county court for your losses on the money claim site (http://www.moneyclaim.gov.uk) although I recommend sending the landlord a letter before action (copy attached) to warn them of your intentions first and allow them 14 days to reimburse you. If they do not reimburse, you are free to issue a claim.
I hope this helps - if you can please accept the answer and give me a 5 star rating (there should be a button at the top of your screen to do this), I can answer follow up Q&A's at no extra charge and Just Answer will credit me for helping you today.
it would help if the council did state the property is uninhabitable but they obviously do not deem the mould a category 1 hazard where they "must" take action. You can make a complaint to the council here:
If you have evidence (photographs of the mould) then you can still pursue this. The council may only intervene in the most serious of cases and if you have an illness as a result of the mould, but the law I mentioned still applies
the risk is he sues for the rent once you leave but you can defend the claim easily on the basis the property was in disrepair. It is easy to defend a small claim, the court sends you the papers to fill out and return.