I am sorry to learn of this matter.
Unfortunately these kinds of issues can be a little complicated. To start, you will want to carefully check the paperwork you signed at the time that you received your care. You want to confirm that the office was responsible for billing your insurance, and not you.
Second, you will want to see if you can submit a bill to your insurance now. Some insurance companies will accept late billing for services as long as the coverage was intact at the time of service. If this is the case, your carrier will pay for the amount due, and deal with this problem for you.
Third, assuming that you were the one responsible for billing your insurance, you can try reaching a settlement agreement with the dentist (paying them part of what they are demanding in full, this is especially true if they offer billing of insurance as a "courtesy" and they didn't inform you - if they informed you sooner, you could have done something about it - their failure to inform you earlier should significantly reduce your liability).
Fourth, assuming that they were the ones responsible for billing your insurance (again under the terms of the service agreement you signed), you can reject their demand for payment and tell them that their negligence in failing to bill acts as a breach of contract and they are barred from collecting the additional amount from you.