Mr Baird said the NSW government is doing everything possible to support Daniel and Benish Khan after their newborn was mistakenly given nitrous oxide instead of oxygen at Bankstown-Lidcombe Hospital in June.
"We will expedite anything we can and certainly that's what we are doing at the moment ... we (want to) take as much stress off as we can," Mr Baird told reporters in Sydney on Tuesday.
Another baby, John Ghanem, died in July in a similar mix-up at the same hospital.
The hospital's general manager and acting director have been stood down and an investigation is underway.
Mr Baird promised there would be accountability for what happened but is continuing to stand by embattled Health Minister Jillian Skinner.
She was taking action to ensure no more tragedies and had his full confidence, the premier said.
"I think the health minister is doing a good job in very difficult circumstances," he said.
"There's no perfect way to respond to this, there really isn't.
"What we have to do though it focus entirely on those families ... we need to provide as much support for them as we possibly can."
A preliminary report pointed to "a series of tragic errors" at the hospital including incorrect installations of gas pipelines, flawed testing and significant clinical and management failures.
The Khans have hired lawyers to handle their compensation claim, saying Amelia has constant seizures and requires a high level of constant care.
"We remain full of hope for Amelia's future but we worry for her and want to do everything we can to make sure she has the best life she possibly can," the couple said in a statement on Tuesday.
"She will always have to live with the consequences of what was done to her at the hospital."
Libby Brookes from law firm Maurice Blackburn is representing Mr and Mrs Khan and said Amelia's injury sustained at the hospital has had a "shattering impact on her parents and extended family".
"Like all parents, they were simply looking forward to welcoming their child safely into the world. Now their lives have been turned completely upside down," Ms Brookes said.
NSW Health Minister Jillian Skinner has faced torrid questioning over her handling of the gas scandal since it emerged in July.
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