Breaking her silence, Kardashian also thanked fans and staff at the Sunrise Hospital in Las Vegas for their support during what she called "an incredibly difficult" week.
Odom, 35, was transferred by helicopter from Las Vegas on Monday to a hospital in Los Angeles, with Kardashian at his side. Kardashian's four-year marriage to the former Los Angles Lakers player was chronicled on the reality TV show "Keeping Up with the Kardashians."
Last week, Odom's prognosis looked grim after several days on life support and in a coma following a reported series of strokes during a stay at a Las Vegas brothel where authorities said he used cocaine and an herbal Viagra supplement.
In a message on her official website, Kardashian said that thanks to the care of hospital staff in Las Vegas "incredible strides have been made" by Odom.
"I am so thankful to family, friends and fans who have sent nothing but prayers and well wishes to Lamar," she said.
A spokeswoman for JaNean Mercer, Odom's aunt, said in a statement that Odom "continues to make miraculous progress, taking a few steps in Los Angeles."
Odom is expected to face months of rehabilitation, according to celebrity media reports. TMZ.com reported he may need a kidney transplant in the near future.
Odom and Kardashian split up in 2013 but their divorce has yet to be finalized.
(Reporting by Jill Serjeant; Editing by Will Dunham)
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