Sorry to hear of the problem.
You can apply to the telecommunications Ombudsman - which is free to use and is quicker than court action.
It is an arbitration scheme meaning the Ombudsman's decision is legally binding if you accept it.
You do have to make a formal complaint to the broadband provider first before you can avail of the Ombudsman though.
The Ombudsman will not help until you have the provider's final decision.
You can allege breach of contract against the provider (you have consumer rights here which have been infringed in my view because you cancelled your service them - any contract can be cancelled in writing - yet they keep taking your money).
Given the circumstances of your case I recommend that you make a formal complaint to the broadband provider and if that does not resolve the matter ask them for a final response letter (a "deadlock" letter).
They will have details of how to make a complaint on their website if you take a look.
Once you have the deadlock letter the next step would be to escalate this to the Communications Ombudsman - they will investigate and liaise with the provider which would hopefully result in a refund of any further monies paid.
They can also order the provider to make a financial award if they have acted poorly.
The service is covered by the Consumer Rights Act 2015 so you have a right to expect a reasonable service - if they do not provide that then the regulator will step in but it also means you can sue for any damages or losses incurred as a result of the provider's conduct.
You can escalate to the Ombudsman here: http://www.ombudsman-services.org/sectors/communications Or by email:***@******.***
If you have lost any money then you could sue the provider but the court may consider that to be premature if you do not go through the Ombudsman first of all.