"We will look for opportunities to avoid this happening again," King said.
He said Bell had closed Sapphire's case and a coroner could not reopen it.
"Someone who wishes for the case to be reopened should apply to the Solicitor-General," King said.
Sapphire died with an alcohol level in her tiny body which was more than six times the legal blood alcohol limit for an adult driver.
Bell's report into the baby's death on January 2, 2017, in Ahipara, was released last Friday with the coroner stressing the importance that breastfeeding mothers should not consume alcohol at any stage.
A toxicology report presented to the coroner showed the amount of alcohol found in blood from Sapphire's heart was 308 milligrams per 100 millilitres of blood.
For comparison the legal blood alcohol limit for a New Zealand driver aged 20 years and over is 50 milligrams per 100 millilitres.
Sapphire and her twin sister, Honey, were delivered by caesarean at North Shore Hospital on November 4, 2016. They were pre-term born at 33 weeks gestation with low birth weights and related medical issues.
Teresa Tua, who was speaking on behalf of the immediate family, has written to the chief coroner questioning why they never received a report and why the coroner carried out her investigation "on papers" in chambers and not in a courtroom where the wider family could have been involved.
Tua said Sapphire's mother and other immediate family found out the report had been released when they saw abusive messages on Facebook.
"It was a shock for the family to learn these findings had been published without receiving a report. They were unaware the media was going to release the findings or the names to the public, nor were they aware of what the findings contained," Tua said.
"According to the report it was done in chambers. The family feels this move by Coroner Bell had disallowed them the right to dispute certain facts."
The family had raised their concerns about the process because they wanted to prevent others suffering the same pain they had.
"In plain words we are outraged," Tua said.
"I urge the chief coroner to look into the matters that deeply troubles our family."
In Bell's report she notes receiving a statement from a Sudden Death in Infants (SUDI) liaison officer and said coroners relied on reports from SUDI which provided detailed information and explanation from the parents as to how the death occurred.
However, in this case the officer was not successful in meeting with the parents. She had two conversations with Sapphire's paternal grandfather, who attempted to facilitate engagement but to no avail.
"This is disappointing given the possible contributing conditions to Sapphire's cause of death and that the parents may have benefited from discussions with SUDI," Bell said.
The cause of death wasn't ascertained, but Bell warned women not to drink while breastfeeding. Contributing factors cited included a dangerous sleeping environment, prematurity, possible septicaemia, suffocation – and acute alcohol intoxication.
Sapphire's mum mostly formula-fed her but said she sometimes breastfed too.
The day before Sapphire died her mum said she consumed a large amount of alcohol.