Ohh I like questions like this.

As above, give the Landlord a bit of time to respond. If has done nothing/informed you of planned action, within a week, tell him you are going to contact the council Environmental Health department.
That should put the fear of God into him/her.
They have powers to make the landlord comply with the law. They also might be able to get out of your contract, should you want to.
Or, buy a cat.

To save my fingers, here is a transcript of an appropriate answer.
As a tenant you have the right to have your accommodation kept in a reasonable state of repair. You also have an obligation to look after the accommodation. The tenancy agreement may give details of both your landlord’s and your responsibilities in carrying out repairs and you should check this.
I would initially suggest that you speak to your Landlord and alert him to the difficulties if you have not already done this.
Local authorities (District Councils and the Northern Ireland Housing Executive in Northern Ireland) have a duty to take action against private landlords, housing association landlords, or, in England and Wales, registered social landlords (including housing associations) if:-
the condition of the property affects the health of the occupier, for example, if it is damp, infested with insects or has a leaking roof; or
the property causes a ‘nuisance’ to people living nearby, for example, if damp is coming into a person’s property from next door, or if rotten windows may fall into the street and injure others.
This duty covers disrepair in common parts of blocks of flats, such as unhygienic rubbish chutes and noisy central heating.
If you think that the condition of the property is either affecting health or causing a nuisance, you should complain to the Environmental Health Department of the local authority (District Council in Northern Ireland). They must investigate and, if appropriate, give your landlord a notice instructing them to carry out the necessary repairs.
If your landlord does not comply with the notice, they could be prosecuted and the local authority (District Council in Northern Ireland) can carry out the repair work itself.
These authorities also have a duty to take action against a private landlord, housing association landlord or, in England and Wales, a registered social landlord (including a housing association) if they consider that housing conditions are not acceptable for people to live in.
The environmental health department of the local authority should order your landlord to sort the problem with the mice infestation. If they are not helping then you should consider making a complaint to one of your local councillors who can put pressure on the council for you.
It is also advisable to put a repair request down in writing. You should keep copies of letters to your landlord.
The other option is to go to the CAB and get them to write (for free) a letter to both the council and your landlord.
Jonathan.