In the video, Bae asks the Swedish diplomat to tell his family that "I have not lost hope and have not given up anything."
But says he is concerned that if his situation isn't resolved soon, it could "drag on" for months longer. He notes that annual U.S.-South Korean military drills due to start later this month may deepen tensions in the region, as they did last year.
State Department spokeswoman Jen Psaki on Sunday expressed disappointment that King's visit was called off and noted North Korea had said it wouldn't use Bae as a "political bargaining chip." It is the second time North Korea has canceled a planned visit by King.
Psaki said that the joint military exercises are "in no way linked to Mr. Bae's case."
North Korea has been urging the South not to take part in the drills -- a call that Seoul and Washington have rejected. Jackson said he hoped a visit would break "that cycle of terror and fear and irritation" between the two countries.
Life in the camp
In the conversation Friday, Bae discusses details of his health problems, as well as the minutiae of life in the labor camp.
He says he is suffering from back pain and neck pain, making the eight hours of manual labor he does each day "very difficult."
"I've been working with my hands a lot," Bae tells the diplomat. "My hands all got numb and sore. I have some cuts."
But he says that he remains "strong mentally and spiritually, and I am trying to stay strong emotionally as well."
Bae tells the diplomat that he has access to books and television at the camp and that the staff there treat him "very fairly."
The TV antenna stopped working for a couple of weeks recently, he says, allowing him to spend "more time with the Lord, with the Bible."
"That was actually a pretty good time for me," Bae says.
Sweden represents U.S. interests in North Korea because the United States has no diplomatic presence in the secretive state.
"We again call on the (Democratic People's Republic of Korea) to grant Bae special amnesty and immediate release as a humanitarian gesture so he may reunite with his family and seek medical care," Psaki said Sunday. "We will continue to work actively to secure Mr. Bae's release."