Kardashian, 31, arrived at about 8 p.m. on Tuesday night, a source says.
"She's terribly worried and distraught over everything. She tried for so long to help Lamar get better, but he just can't get rid of his demons," the source says. "She still cares so much about him and wishes desperately that she could help him. They talk on the phone and communicate, but half the time she can't reach him. She's put so much energy and heart into trying to be there for him through all his troubles."
Jenner, 59, and Kardashian West, 34, were photographed wearing all black while entering and leaving the Las Vegas hospital where Odom, 35, was transported Tuesday. As of late Tuesday night, Odom was breathing but still on life support, the source adds.
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Another source says the famous family is rallying around Lamar in his time of need.
"Everyone has dropped everything to get there. There are a lot of calls and texts going back and forth as everyone is figuring out how they can help and be there for each other," the source says. "They are asking their close friends to pray for Lamar, for healing and strength. There is confusion about his condition, and they don't know the extent of his injuries, but I'll tell you this: they wouldn't just drop everything unless it was serious. The family has been told that this isn't just a minor thing; that it was extremely serious."
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Odom was discovered unconscious at Dennis Hof's Love Ranch in Nevada on Tuesday afternoon.
"We don't know why he seemed a little sad, but he had been in good health and high spirits, laughing and giggling," Hof told PEOPLE of the former basketball star's behavior since he checked in Saturday.
Kardashian married Odom in 2009 after a whirlwind romance. They separated in December 2013 because of his substance abuse, but Hof said he "wasn't taking any illicit drugs" besides herbal Viagra at the ranch.
"Everyone is very upset," the source close to the Kardashian family says. "They know that Lamar has his demons, but the fact is that he was a very important part of Khloé's life for so long, and she's very upset about this. They all are."
"Khloé was very hurt by Lamar, but she never got to the point where she hated him or was indifferent to him," the source adds. "She wanted him, and still wants him, to pull it together and live a happy life. She doesn't wish him harm. Even though they are not together anymore, I know that she cares about him and is devastated by this."
Reporting by ELIZABETH LEONARD, STEVE HELLING and AILI NAHAS