You need to find out if she is occupying under a tenancy agreement which still has some of the fixed term remaining, or if the fixed term of the tenancy agreement has expired and she is now occupying the property under a statutory periodic tenancy in which the tenancy continues under the same terms but from month to month.
You should formally write to the the landlord specifying the disrepair, 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 pay to have the repairs made and will seek to claim the expense from him and are prepared to make an application to court if necessary.
If the landlord does not make the repair you can pay to have it done and then write formally to him requesting the payment of the cost, again within a reasonable time). If he does not pay you can issue a claim for the money yourself through Her Majesty Courts Service's online service: http://www.moneyclaim.gov.uk/
The difficulty is that if she is occupying under a statutory periodic tenancy then upon receiving the letter the landlord may just serve a s21 notice giving her two months notice to vacate the property and therefore the benefit/stress of repairing the boiler and claiming the money from him would be reduced because she will have to leave the property a relatively short amount of time after arranging the repair.
My advice would be to write to the landlord in the above way. Find out if the tenancy is still in term or not and then decide if making the temporary repair is worth it in terms of cost/benefit. It is not difficult to moneyclaim him in the above way, but for this to be viable the landlord would need to have assets/income against which judgement could be enforced.
If you mother put down a deposit upon entering the tenancy and the tenancy was an assured shorthold tenancy (which almost all are if the rent is less than £25k pa) then the landlord must have placed the deposit in a tenancy deposit scheme. If he has not then she can sue for the the return of the whole of the deposit and a fine payable of three times the deposit amount from the landlord to her. You can keep this in the back pocket to exert some leverage if he has not so placed the deposit.
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