By law, landlords must ensure the property is in good repair under S.11 of the Landlord and Tenant Act 1985.
If they do not, they face prosecution by the local council and can be fined. It would also be breach of contract on their part and you would be within your rights to terminate the tenancy agreement. Damp is a health hazard given respiratory problems with mould spores.
The Local Authority Environmental Health Officers have powers under the Housing Act 2004. They have a duty to ensure that properties in their area are in a habitable condition, and will serve improvement notices on landlords of properties which are assessed, under the new Housing Health and Safety Rating System as having ‘category 1 hazards’.
Similar powers are available to Local Authorities under the Environmental Protection Act 1990 if the property is considered a threat to public health.
I would therefore urge you contact your local council (the housing department) in the first instance to report the matter and ask them if they can send someone out to do an emergency repair - if so, they will do the work and they will bill the landlord themselves. If for some reason the council cannot help, you would either have the option of paying it yourself and withholding rent (not applicable here as you have already paid), or to instruct a builder to come out and fix the problem and you then pursue the landlord for the money.
If the landlord does not pay or if they refuse for whatever reason, you can sue them 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 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.
If you want me to fine the number at the council, just let me know which area you live in and will find the right number for you to contact.
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