If you send them a letter before action first - they will want to avoid court action and I would think the letter will work. It needs to give them 14 days to pay. You can post it and also send by email. In the meantime, I recommend you make a complaint to them to speed matters up.
You can avail of the Ombudsman scheme here - which should resolve the dispute. It is free to use - you have to first make a complaint to the energy company, wait for their final response and then go to the Ombudsman.
I recommend you should now make a formal complaint and if that does not resolve the matter ask them for a "deadlock letter", which is a letter giving their final response.
The complaint to EON Energy can be made here: https://www.eonenergy.com/contact/complaints.html
Once you have the deadlock letter the next step would be to escalate this to the Energy Ombudsman - they will investigate and liaise with the energy company which would hopefully result in a payment of what is owed. They can order the energy company to make a financial award for inconvenience 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 which is carried out with care and skill - if you do not receive this then the regulator will step in but it also means you can sue for any damages or losses incurred as a result of the energy company's conduct.
Once you have EON's final response you can complain to the Energy Ombudsman by email here: *****@******.***
Or via their website here: www.ombudsman-services.org/energy-complaint.html