Arna's father, Regan Hopkins, suffered severe burns as he tried to put out the flames.
His wife, Penny, is with their little daughter in Auckland.
The couple have two other children, a 5-year-old boy and a 6-year-old girl.
Mr Hopkins' brother, Kane Hopkins, said the family were devastated to have suffered another tragedy so soon after Mila's death.
"This is doubly heartbreaking and they are again having to face the situation that no parent really wants or should ever face," he said.
The family had been lighting candles in Mila's memory and it was believed that Arna had climbed up to get a box of matches to light one herself.
"She had to climb into a pantry and she lit a candle - which they had been doing as part of a grieving process for the family - and somehow, horribly, that candle set her dress on fire," Kane Hopkins said.
There was a lesson for other parents to ensure their children were wearing clothes that would not catch fire, he said. Arna had been wearing a 100 per cent cotton dress at the time.
"We are buying clothes that are so flammable ... that children are at risk."
Middlemore Hospital spokeswoman Lauren Young said Arna had surgery yesterday and was later in a critical but stable condition.
Family friend Jo Hanson is raising funds to help the Hopkins.
"This family has been through so much. Because Mila had ongoing health issues, Penny spent long periods of time in the Starship away from the rest of the family and now, with Arna's injuries, they are facing weeks, if not months, of separation.
"The least we could do for them is try to raise some money so they can be together as much as possible."
Donate to: Support For Arna, National Bank 06-0689-0298651-00
