The research work shown in the video was published in New England Journal of Medicine
Two girls, unidentified in the paper, in the small clinical trial were given the same treatment. One was successfully cleared with cancer cells in 11 months (must be the girl in the video). The other had a relapse 2 months after the treatment. The cancer cells changed their facade, and the killer immune cells couldn't target them any more. I am very happy that this approach has cured the little girl, but there is obviously a long way to go to make it successful to every patient with leukemia.
The virus they used is lentivirus, which belongs to the same virus family with HIV. The researcher engineered a small piece of the HIV into the lentivirus. So it was not HIV they used.
The researchers did not inject the virus into the little girl. Instead, the immune cells were taken out and treated with the engineered virus, which helped carried a gene into the immune cells. The engineered virus would't replicate any more, but the gene would be expressed and produce proteins that recognized the cancer cells. These killer immune cells were injected back into the little girl to fight cancer.
I am all for the public awareness and potential fundings brought by the video, but it should not be done at the price of sending out scientific misconceptions.