Nguyen's husband, Khoa Le, 52, is in a medically induced coma at a local hospital. He reportedly suffered burns over 70 percent of his body.
Le, Nguyen and their son emigrated to the United States from Vietnam, according to Councilman Nguyen.
A representative for the manager and owner of the building says the family had lived in the duplex for at least 10 years and emphasized the apartment was safe and had been inspected frequently by city officials.
"Everything was up to code," said Georgia Tsaboukos, the niece of Afroditi Andrews, who owns the building. All four units in the building had smoke alarms, Tsaboukos said.
"I feel for the whole family," Tsaboukos said.
The San Jose Fire Department's Bureau of Fire Prevention has yet to release its inspection records of the South Ninth Street building but Capt. Vega said the last check of the property, in June 2017, gave it a clean bill of health.
However, it's unclear if the fire crew that inspected the building actually entered the unit where the blaze took place.
In San Jose, as in other California cities, firefighters are tasked with inspecting apartment buildings once a year. Those crews often check the common areas of the structures for fire safety violations.
It's common, though, for firefighters not to get inside individual units because residents may not be home at the time.
"We're certainly not going to be breaking down any doors," Vega said. "If ... everything looks good from the outside, we may not be overly suspicious that things won't look good on the inside."
Some areas of the city may get less attention than others, Vega said.
"The busiest fire stations have the most apartments, so they have trouble keeping up with the workload or at least they can," he said.
The fire has prompted Councilman Nguyen to call for improvements in making San Jose's low-income and older housing stock more fire-safe.
"In poor communities, especially immigrant folks, a lot more people, due to the shortage of housing, are staying in more cramped rooms," Nguyen said.
Saturday's blaze marks the second fatal residential fire in San Jose in the last three months.
A fire Aug. 15 at the Golden Wheel Mobile Home Park, a largely Vietnamese-American community near the Berryessa neighborhood, killed three people.