But steering families to hotel rooms instead — another stopgap solution — costs even more, and flouts another promise by the mayor: to phase out the use of hotel rooms as shelter for homeless people.  “The cost of housing in this town continues to go up, wages are not going up anywhere near the same with it, more people are becoming homeless,” Mr. de Blasio told reporters before an event in Queens on Thursday.  “We’re trying everything we know to keep them out of shelter,” he added, “to find any other way to get them someplace else better or, if they get in shelter, to get them out of shelter quick and back to some kind of housing. But this is a constant flow of people we’re trying to support and address.”  The sisters killed on Wednesday, Ibanez Ambrose, 2, and Scylee Vayoh Ambrose, 1, died from overheating and thermal injuries from exposure to steam, said Julie Bolcer, a spokeswoman for the city Medical Examiner’s Office, which ruled the deaths an accident.  The episode occurred in an apartment on Hunts Point Avenue where the girls and their parents had been placed by the Department of Homeless Services. The girls were sleeping in a bedroom — one in a playpen, the other in a bed — when the valve on a radiator apparently detached, spewing steam into the room, the police said. The girls’ father was in another room at the time.  “What we know so far suggests an extraordinary and unprecedented accident, something that we — no one that I’ve talked to so far in any agency has ever seen anything like this.” Mr. de Blasio said.  It was unclear whether conditions at the building — whose owner, Moshe Piller, was ranked No. 4 on a list of the city’s 100 worst landlords prepared by the public advocate’s office in 2015 — had led directly to the toddlers’ deaths.  The mayor, who said the inquiry into the deaths was continuing, visited the apartment on Wednesday, accompanied by Steven Banks, commissioner of the Department of Social Services.  “I looked at this apartment personally very, very carefully,” Mr. de Blasio said. “There were not evident health and safety dangers in this apartment.”  The girls’ father, Peter Ambrose, said in a telephone interview on Thursday that he and his wife happened to have been awakened around 6 a.m. on Wednesday, when a valve on a different radiator — in the living room, where they were sleeping — “blew off” and steam poured out.  Mr. Ambrose said that the valve was extremely hot and that he had to use an oven mitt to pick it up and put it back on the radiator.  The girls were asleep in the adjacent bedroom, with the door closed, he said. Ms. Ambrose then left to run errands, he said. “She gave me a kiss and told me to keep an eye on the girls,” he said, adding that he had gone back to sleep.  Around 11 a.m., he said, Ms. Ambrose returned. “She opened the door of the room,” Mr. Ambrose said, his voice quavering. He was unable to speak further.  In a later interview, Chad Boonswang, a lawyer for the family, said, “When the mother went to rouse them, she opened the door and immediately saw all the steam in the room.”  Ms. Ambrose picked up one of the girls, who was blistered and purple, and ran outside screaming to call 911; her husband picked up the second girl, whose skin was also swollen and discolored by burns, Mr. Boonswang said.  The valve on the radiator in the bedroom had blown off, he said. “There was steam pouring in the room for God knows how long,” Mr. Boonswang said.  Mr. Banks said that an inspector from the Department of Homeless Services had visited the apartment on Monday for a routine inspection and had reported finding no problems. “They didn’t notice anything untoward,” Mr. Banks said.  A Fire Department official describes the response to the fire at a Bronx building where two toddlers were killed on Wednesday.  Mr. Banks said that what he saw in the apartment reflected a loving family.  “There were toys, there were pictures of the parents together, a picture of the parents together with their children,” he said. “Drawings that had been made, obviously, by a parent from the children to the other parent. Valentine’s Day, Happy Birthday, expressions of love for one another. It was just heartbreaking.”  Mr. Banks, citing a desire to protect the family’s privacy, declined to say how long they had lived in the apartment or to provide other details about their contacts with their authorities.  The family did have some contact with the authorities, according to a police official who spoke on the condition of anonymity because the investigation is continuing. Employees of the Administration for Children’s Services questioned the couple last month because of a report that Ms. Ambrose had been seen playing a guitar and panhandling with the two girls at Grand Central Terminal, the police official said. Officials also visited the family’s apartment in August after the couple got into a loud argument, the police official said.  Mr. Ambrose said the couple had moved to New York from a small town in Maine because they wanted to raise their daughters in a big city. “We wanted to give them something we didn’t have when we grew up,” he said.  He said he had found occasional work as a tattoo artist and as a house painter, and that his wife was a musician and had recently gotten work as a security guard.  “It was a little harder to get it all going,” he said. “I didn’t research the prices. It was more like one of those things you think you can do, dream you can do. It was more of a struggle than we anticipated.”  The family’s struggle illustrates one of the many paths to homelessness in New York, and why the city is wrestling with a growing shelter population. Mr. Banks said that in January, when the decision was made to phase out cluster sites, there were about 3,600 such apartments. Since then, he said, about 450 have been eliminated, leaving about 3,200, with about 12,000 people living in them.  “The tragedy of last night highlights the urgency of getting out of clusters, the urgency of getting out of commercial hotels, and getting purpose-built shelters in place around the city,” Mr. Banks said. He said that nine new permanent shelters had recently opened and that a plan was being developed to build more.  But Mr. Banks said the city could not simply abandon cluster sites without finding other places for the families living in them.  Some homeless people have been put into hotel rooms paid for by the city instead of being placed in cluster apartments. But hotel rooms are more than twice as expensive as cluster apartments, and present other problems, including, in many cases, the absence of kitchens.  Mr. de Blasio vowed, after a homeless mother and two of her children were murdered in February in a Staten Island hotel where they had been placed by the city, to rapidly reduce the city’s use of hotels as a form of shelter.  Instead, the opposite happened. In January, there were 2,449 homeless people staying in several hundred hotel rooms. On Nov. 30, there were 7,289 homeless people staying in more than 3,000 hotel rooms, according to the Department of Homeless Services.  An earlier version of this article referred incorrectly to where New York has sheltered some homeless residents. Some homeless people have been placed in hotel rooms paid for by the city instead of being placed in cluster apartments; the city has not moved people from cluster apartments to hotels.  Eli Rosenberg, Nate Schweber and Ashley Southall contributed reporting.  A version of this article appears in print on December 9, 2016, on page A1 of the New York edition with the headline: Sisters’ Deaths Reveal Strains in City System. Order Reprints| Today's Paper|Subscribe
The tragedy underscored the challenges Mayor Bill de Blasio faces in addressing an issue where even the potential solutions can create further problems.