“She’s in bed 99 percent of the time. She can’t do anything,” Marzia Waziri said in a phone interview from the city of Aarhus, where she lives in an apartment with her mother and runs a small grocery store.

Ms. Waziri’s condition has worsened in the two years since The New York Times first reported on her case. Once able to walk around her daughter’s apartment, she chokes because she forgets to swallow her food, wears a diaper and barely recognizes family members, her daughter said. In addition, she has suffered blood clots and is struggling with diabetes and asthma, Marzia Waziri said.

She said she would defy the order and refuse to deliver her mother to the deportation center. “We’re not going there,” she said. “They will have to come and get her.”

Leif Randeris of Danish Immigrant Counseling, who is working on the case on the family’s behalf, said the Ministry of Immigration and Integration had sent out 15 to 20 requests for more or updated information in the current appeal, which was opened in 2016. Earlier this year, when the authorities made yet another request, Marzia Waziri said that she had reached her limit.

“They keep on and on and on every three months asking for new information,” she said. “I don’t want to mail more paper. It’s been going on for two years and I’m thinking of this every day. I can’t gather more proof.”