Hello I am a solicitor with 20 years experience. I will try to help you with this.
It seems clear that your dentist owes you this money and should repay it immediately. You can follow this link to get a model of a complaint letter to your dentist from provided by the General Dental Council. Fill it in and tell them that if you do not get a refund within 14 days you will report them to the General Dental Council and issue proceedings in the small claims court for recovery of the money.
The small claims court is designed to be usable without solicitors for either the claimant or the defendant. This is the line to the government site providing guidance on how to do it https://www.gov.uk/make-court-claim-for-money/overview.
Hopefully making the threat will be enough. You should include proof of the payments in the letter you send to the dentist.
Shahla Imani, dds & associates
Dear Madam, I have been given expert legal advice from the ombudsman representing the dental profession who advises me after examining the letter of confirmation from the Halifax Clearing Centre that my first cheque for £232.52p was indeed credited to your Bright on Smiles account on the 06/08/2014 and therefore you should return that amount back to me. Despite my numerous attempts to reason with you, you still refuse to return my money or to supply proof that it wasn't credited to your account on the 6th August 2014. Under the difficult circumstances you are creating we have no alternative other than to demand a refund within 14 days and to report you to the General Dental Council. Your immediate reply and explanation by return please
That seems fine but you should include a copy of proof of the transaction (eg a print out of your statement with cnfidatial information removed). You have also not threatened court action, I assume that is deliberate.