Sorry to hear of the issue.
The lack of a boiler is a serious breach of the landlord and tenant law - landlords must provide heating and hot running water to the property.
It is classed as an emergency and therefore needs to be fixed within 24 hours. They also have to ensure the property is in good repair under Section 11 to 16 of the Landlord and Tenant Act 1985.
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 now 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 lack of heating would put the landlord in breach of this law. The local council's housing department could be contacted if the landlord and agency fail to repair.
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 terminate then they could sue and you may have to defend the claim in a civil court.
You may want to ask a no win no fee lawyer to help - they will not charge you up front and if any damages are recovered, they would take a set percentage.
If you have to sue the landlord, you may want to use a no win no fee lawyer - there is nothing to pay up front and you are also protected from legal costs if you lose.
Here are a few choices for you if you would like to use a lawyer :
1. Knights Law, tel: 01204 655614, email:***@******.***
2. Stephensons Solicitors, tel: 0161(###) ###-####
3. Duncan Lewis, tel: 0333(###) ###-####