My view is that you are due the money back from when you put the landlord on notice of the problem. You could either withhold rent to cover the shortfall or issue a small claim - either way, you need the money back.
To summarise the law, landlords must ensure the property is in good repair under Section 11 of the Landlord and Tenant Act 1985. Meaning the boiler should not be faulty. A faulty boiler can also be a health hazard (carbon monoxide issues being the most serious) so it would be highly recommended to get it checked out, either from the council or a heating engineer.
Further, a recent law came in to force to provide more protection to tenants, which is the Homes (Fitness for Human Habitation) Act 2018 is not in force - the house or flat needs to be healthy, safe and fit for human habitation (free from things that could cause serious harm). So a faulty boiler would put the landlord in breach of this law.
If they do not, they face prosecution by the local council and can be fined. It would also be a breach of contract on their part. The only situations where a tenancy can be ended are, you finish the tenancy term or there is a break clause meaning you can end the tenancy part way through. Or the landlord agrees to you leaving early - usually if another tenant can be found. You could terminate but you would need evidence of repeated requests to the landlord to fix and they do nothing. If you terminated then they could sue and you may have to defend the claim in a civil court.
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’.
I would therefore recommend that 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.
You can reach yours here: https://www.bromley.gov.uk/forms/form/437/en/report_disrepair_in_your_private_rented_property
If for some reason the council cannot help, you would either have the option of paying it yourself and withholding rent or to instruct a heating engineer to come out and fix the problem and you then pursue the landlord for the money for the repair, plus the higher bill amount.
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 (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.