This is exactly the type of factual dispute that is the reason that they build court houses. Document everything. Of course you have a copy of the written contract. Take pictures of all of the work that shows defective workmanship. You need photocopies, or the originals, of all payments or receipts for payments. Then file a claim either in the local small claims court or the district court depending on how much in damages that you seek. The court can not force the contractor to do anything, this would require injunctive relief. All the court can do is award you money damages. Finally, if this is not entirely clear, and if the court is going to be confused by the arguments on both sides, it would not hurt for you to have another contractor come in to the court to testify in your favor. (No letters or affidavits, court requires actual live testimony. Unless you are in small claims court where the judge may or may not bend the rules a bit.)
This answer is provided for informational purposes only. Actual legal advice can only be provided in an office consultation by an attorney licensed in your jurisdiction, with experience in the area of law in which your concern lies.