As in all cases of personal injury, you should consult a specialist personal injury lawyer who has knowledge and expertise in the area of medical negligence. They will be able to assess you case and let you know whether your dentist was negligent. You will need to prove that you suffered an injury or accident at the hands of your dentist and that they were negligent, in other words they failed in their duty of care and did not provide you with the care that another dentist would have.
It is imperative that you consult a personal injury lawyer as soon as possible after your accident so that you can start the claim process and so that your solicitor can gather the evidence needed to support your case. Your solicitor should be able to work on a no win no fee basis, also called a conditional agreement.