Praise be to Allah.
Firstly: If the dentist makes an error, he is liable, even if he is skilled, but there is no sin on him in that case. If he is not skilled, then he is sinning and is liable.
Ibn Qudaamah (may Allah have mercy on him) said: If (a doctor or surgeon) is very skilled but makes a mistake – such as if he goes too far in cutting the foreskin (when doing a circumcision) and damages the tip of the penis or part of it, or cuts somewhere other than the place that is to be cut; or if he cuts into a cyst and goes too far; or cuts with an instrument such as an instrument that causes too much pain; or he cuts at a time when cutting is not appropriate; and the like – in all these cases, he is liable, because this is causing damage, and no scholar differed concerning the fact that he is liable, whether he did that deliberately or by mistake. This is like causing damage to wealth or property. Moreover, this is a haraam action, so he is liable for it, like one who cuts deliberately. The same ruling applies to practitioners of hijaamah (cupping), veterinarians, and those who carry out legal retributions (qisaas) and cut of the hands of thieves. This is the view of ash-Shaafa‘i and as-haab ar-ra’y, and we do not know of any difference of scholarly opinion concerning that.
End quote from al-Mughni (6/133).
Regarding cases of medical liability, please see the answer to question no. 114047; that includes doctors who make errors in prescribing medicine, leading to loss of limb or life.
What appears to be the case from your question is that the dentist made a mistake in what she was doing; her mistake may have been one of two things:
Firstly: the tooth did not need to be drilled so deeply that it exposed the nerve and made what you call a hole that was too deep and led to nerve damage, which led to extra costs for the special type of filling required for that, then later on putting a crown on the tooth.
Secondly: the tooth needed this type of drilling, but she made a mistake by putting the filling directly on top of the nerve, or she made a mistake by using a regular type of filling that is not used in the event of nerve damage.
Whatever the case, reference concerning that should be made to experts in that field. The matter should be referred to two trustworthy dentists, so that they can determine the nature of the dentist’s mistake and whether she caused damage to the tooth, or made a mistake with regard to using the appropriate materials.
She is liable for the damage that she caused, so she should bear the costs of treating what she damaged. That is also to be determined by experienced doctors.
If a tooth is removed completely by mistake, then the liability for that is five camels.
If it was not removed, but the root was damaged, then the liability is to be determined by artbitrators, who should be experts in the field.
With regard to the dentist’s fees, if she did not treat the tooth, rather she damaged it, then she does not deserve any payment.
Dr. Huwaymil ‘Awjaan said in his paper, Damaan at-Tabeeb fi’sh-Sharee‘ah al-Islamiyyah wa’l-Qaanoon:
If a doctor makes a mistake in his work and his mistake causes harm to the patient, he does not deserve the fee dictated by the contract; rather what he took from the patient should be taken back from him, because what he did did not benefit the patient.
In addition to that, he is also liable for other medical costs that result from attempts to relieve the harm suffered by the patient.
The Messenger (blessings and peace of Allah be upon him) said: “There should be neither harm nor reciprocating harm.” This is one of the basic principles of Islamic teaching, because the one who causes harm must bear the consequences and outcomes of that. The Prophet (blessings and peace of Allah be upon him) forbade responding to harm in kind, therefore the correct way is to assume liability for harm.
It says in ad-Durr al-Mukhtaar: The one who was wronged has the right to take back from the one who wronged him what he spent on the cost of treatment and medical fees.
It should be understood that in cases of disputes, a fatwa is not sufficient, because it is essential to hear both sides and find out the facts of the matter. So you should refer your case to scholars in your country, after obtaining medical reports from the two dentists who are appointed as arbitrators.
And Allah knows best.