In the late summer of 2016, as the city scrambled to reinspect apartments for lead paint hazards, Mikaila’s blood lead level hit 37 micrograms per deciliter, nearly eight times the amount that prompts Health Department action.

After the Housing Authority told the Health Department that the lead could not have come from its apartments, Mikaila’s family said she was still not herself, by turns lethargic and hyperactive. Occasionally, said her grandmother, Ordeen Broomes, she wailed with discomfort. A third blood test in late September 2016 showed she still had very high levels of lead.

So the Health Department returned to both apartments and again found lead, according to city records, this time in dust on the floor. At this point, the Housing Authority relented. Workers came with a bucket of cleanser and a special vacuum to suck up the dust.

But no one looked for the source of the lead-riddled dust, according to city records reviewed by The Times. The Housing Authority declined to comment on Mikaila’s case, citing the pending litigation.

Mikaila, now 5 and a kindergartner, has not required any special attention at school, her mother said. Still, said Max Costa, a professor and chairman of environmental medicine at New York University School of Medicine, her experience is “going to totally affect her life, and there’s no way you can reverse it.” The family’s observations are consistent with those effects, Mr. Costa said.

Ms. Broomes, who works for the Parks Department, wants to get her family out of public housing. But it is a struggle.

On a recent evening, she sat at her dining room table holding her head in her hands. A cockroach fell from a kitchen cabinet. Another climbed the wall.

About a year after Mikaila tested positive for lead, maintenance workers painted, patched over a large hole in the wall and laid new tiles on top of her crumbling linoleum floor, Ms. Broomes said. Problems persisted, she said, but saving money for a private apartment or a house was difficult.

As she spoke, Mikaila, sitting beside her, arched her eyebrows at the thought of a house.

“I want stairs for my room,” Mikaila said. “I want stairs so I can go up the stairs so I can go to my room. I want to get a back garden and I want to plant some seeds.”