When a person dies without a Will in NYS their heirs must usually go through what's termed an Administration which means a finalization of their Estate. In an Administration, and Administrator is appointed to "marshal" all the remaining property of the estate and then distribute it according to certain rules set down by the State.
If only children are left, then they will get the remaining assets equally AFTER all outstanding bills, taxes and expenses of the estate are paid. In this case, the medical bills are the primary issue. They would normally have to be paid out of remaining assets.
If any assets are jointly owned, then the joint owner gets those assets by operation of law and it doesn't pass through her estate. If those are all the assets that existed in her sole name at the time of her death, then the medical bills may go unpaid and that's the end of them.
As for a certain amount, usually a small estate is anything under $60,000. and that normally means you don't have to go through a complete large estate proceedings and can do it with certain simplified paperwork which you can get from the local county Surrogates Court clerk's office.
You can utilize a Small Estate Proceeding to appoint a Voluntary Administrator to handle the estate if the estate consists purely of personal property (no real estate) and if the estate is valued at $30,000 or less (Art. 13 of the Surrogate's Court Procedure Act). This requires you to complete a Small Estate Affidavit, which is a relatively simple form that starts a relatively short court proceeding.
Jointly held property is not part of the estate. Life Insurance policies and other accounts with a designated beneficiary are also not part of the estate.