Thanks for your question. I will try to help.
I'm afraid you will be bound by the full term of the tenancy agreement unless there is a break clause in it entitling you terminate at a certain point during the term or the property is unfit for human habitation (which would also entitle you to terminate as it would be what is called a repudiatory breach of contract).
If the damp is sufficiently severe so that no normal and reasonable person could or should live in the property then it would be a repudiatory breach of contract. If the damp is not sufficiently severe then it would not entitle you to terminate the contract.
You should be aware that it is unlikely whatever you do the landlord is likely to dispute it, so it may ultimately end up with litigation.
You should formally write to the landlord specifying the disrepair that you notified them of previously and noting when you told them about it (and that they have not corrected it despite saying they will do so, , making a list of the reasonable repair required and ask that he make those repairs within a reasonable time (eg. 14 days). State that if the landlord does not make the repair within that time you will consider it a repudiatory breach of contract which entitles you to terminate.
If he repairs it then you will have to see out your contract. If he does not then I would strongly suggest contacting the environmental health department of the local authority and getting them to make an inspection. If they declare that the damp is severe then they may serve a notice on the landlord requiring repair. You should also get them to state in writing (if you can) that they consider it unfit for human habitation. If they do this then you may consider seeing a local solicitor and getting them to write to the landlord stating that you consider it a repudiatory breach entitling you to terminate and then thereafter vacating.
If environmental health do not declare the damp to be severe then your best bet is to get the landlord to do the repairs, but if he does not then to pay for the repairs yourself and then seek to recover the cost from him by suing him with evidence of the original damp, your efforts to get him to repair and evidence of what repairs have cost.
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