An attempt to rescue Kayla Mueller and other American hostages occurred in a July 4 raid previously disclosed by the Pentagon.
U.S. special forces commandos conducted a raid in the predawn hours on a prison in Islamic State's Syrian stronghold of Raqqah.
But the mission was unsuccessful because hostages -- which included Mueller as well as American journalists James Foley and Steven Sotloff -- had already been moved and weren't there at the time.
"We never stopped trying to get her," said a defense official, who wasn't authorized to speak publicly on the matter. "We never lost that focus."
President Obama confirmed for the first time Tuesday that Mueller was one of the hostages Delta Force commandos attempted to rescue in a raid on an oil refinery facility in northern Syria in summer 2014. The two dozen commandos arrived a day or two after the hostages had been moved from the site, Obama said.
"I deployed an entire operation -- at significant risk -- to rescue not only her but the other individuals who had been held, and probably missed them by a day or two," Obama said during an interview with BuzzFeed News.
Obama said the U.S. has a commitment to devote "enormous" resources to free hostages anywhere in the world.
Relatives of Mueller, who was abducted while volunteering in Syria, announced Tuesday that they had received confirmation that she is dead.
Islamic State released a video last week claiming that Mueller, 26, was killed in an air attack by coalition forces against the militants in Syria.
U.S. intelligence officials believe that Mueller is dead but don't accept the militants' account of how she was killed, White House Press Secretary Josh Earnest said in Washington on Tuesday.
There was no evidence civilians were in the area that was targeted by Jordan on Friday, Earnest said.
The Pentagon confirmed that Jordanian warplanes bombed a weapons storage facility Friday, but there was no indication that Mueller was there at the time, said Rear Adm. John Kirby, a Pentagon spokesman.
"We do not know the circumstances surrounding her death," he said. "U.S. intelligence had no evidence that there were any civilians at the location of this weapons storage facility that was struck by Jordanian F-16s."
The building, a facility near the Islamic State stronghold of Raqqah, was hit Friday and at least twice before, he said.
Regardless of how Mueller died, he said, her Islamic State captors were to blame. "They are responsible for her safety and well-being," Earnest said.
Mueller's family members released a statement saying that they had received confirmation of her death. They did not disclose how they learned the information.
However, a source close to the family who requested anonymity because of the sensitive nature of the communications said a private message containing additional information from the captors was sent to Mueller's parents over the weekend. Once the statement was authenticated by the intelligence community, the family concluded that she was dead, the source said.
"We are heartbroken to share that we've received confirmation that Kayla Jean Mueller has lost her life," her parents, Carl and Marsha Mueller, and her brother, Eric, said in the statement.
U.S. officials are aware of at least one other American being held in the region, Earnest said. He did not identify the individual, or the group holding the captive. But he may have been referring to Austin Tice, an American journalist and former Marine who disappeared in Syria in August 2012.
It is unclear who is holding Tice, but his parents, Debra and Marc Tice, have said that they don't believe he is in the hands of Islamic State militants.
"Our hearts go out to the Mueller family and everyone who loves Kayla," the family said in a statement posted on a website that aims to raise awareness about Austin Tice's captivity. "We have no way of understanding what they are going through, and we wish them all possible comfort."
In Washington, the White House released a statement of condolence from President Obama.
"On behalf of the American people, Michelle and I convey our deepest condolences to Kayla's family -- her parents, Marsha and Carl, and her brother Eric and his family -- and all of those who loved Kayla dearly," the statement said. "At this time of unimaginable suffering, the country shares in their grief."
Mueller had a long history of volunteering to help women and children, having worked for aid groups in Arizona before setting out for other countries, including India and Turkey.
In August 2013, she was abducted in the Syrian city of Aleppo as she left a hospital operated by Doctors Without Borders and was being held in Raqqah, a militant stronghold in northeastern Syria.
The family statement also included letters Mueller wrote to her family, including one on her father's birthday in 2011.
"Some people find God in church. Some people find God in nature. Some people find God in love," she wrote. "I find God in suffering. I've known for some time what my life's work is, using my hands as tools to relieve suffering."
After attending Northern Arizona University, she worked with aid groups in northern India, Israel and the Palestinian territories. She returned to Arizona in 2011, where she worked at an HIV/AIDS clinic and volunteered at a women's shelter. Late that year, she moved to southeastern France and worked as an au pair while learning French in preparation for a planned move to Africa.
But the plight of families fleeing the violence in war-torn Syria drew her to Turkey in December 2012. She worked with the aid groups Support to Life and the Danish Refugee Council, assisting women and children who crossed into Turkey as refugees.
She also made some trips into Syria to help reconnect family members separated by the fighting. Her trips into the country took her to Aleppo, where she was eventually kidnapped.
In correspondence to her family, which she slipped to other detainees who were being freed, Mueller was contrite and seemed to try to assuage her family's worry. She told them she was being treated well and was not in harm's way.
"If you could say I have 'suffered' at all throughout this whole experience it is only in knowing how much suffering I have put you all through; I will never ask you to forgive me as I do not deserve forgiveness," Mueller wrote in a letter the family said it received in the spring of last year.
She said in the letter that she was willing to wait for her freedom if it meant that her family would be absolved of the burden of negotiating her freedom.
"I DO NOT want the negotiations for my release to be your duty, if there is any other option take it, even if it takes more time," she wrote. "This should never have become your burden."
Her family described her as relentlessly optimistic. Even while held hostage, Mueller said she found reasons to be grateful.
"By God and by your prayers I have felt tenderly cradled in freefall. I have been shown in darkness, light [and] have learned that even in prison, one can be free," she wrote. "I am grateful. I have come to see that there is good in every situation, sometimes we just have to look for it."
Times staff writers Duara and Hennigan reported from Tucson and Washington, respectively. Staff writers Christi Parsons in Washington and Alexandra Zavis in Los Angeles contributed to this report.