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Kimmich made headlines in October when he explained he was not willing to receive his first dose due to what he perceived as a lack of research into possible long-term effects.
The 26 year-old has not featured for Bayern since 6 November after twice being forced to isolate due to being a close contact of someone who tested positive for coronavirus.
He was then infected with COVID-19 himself and was last week ruled out until 2022 due to an issue with his lungs.
Kimmich confirmed in an interview with ZDF that he is now in a position where he feels ready to get vaccinated.
Nagelsmann, who was himself struck down by the illness, is pleased with Kimmich's announcement but insisted he did not sway the midfielder's decision.
"I think it's good that he faced the media. A lot has been reported about him in the last few weeks," Nagelsmann said. "I gave him my opinion, but in the end his conviction was a result of talks with other people at the club. I rather try to work on the pitch."
Nagelsmann, who has been regularly questioned on the topic over the past two months, added: "I think it's funny I'm still being asked!
"When the year 2021 comes to an end, I hope that we will be able to deal with the pandemic subject more briefly because we can master the situation."
Bayern will be without Kimmich for its final two Bundesliga games of the year, while Jamal Musiala is also doubtful to face Stuttgart on Wednesday (AEDT) and Wolfsburg next weekend.
Musiala has scored in Bayern's last two games, including the decisive strike to see off Mainz 2-1, but a broken hand could keep him out of action until 2022.
"That depends on the pain," Nagelsmann said ahead of the trip to Stuttgart. "He has a splint and could theoretically play, but it's a pain issue.
"We'll see how he moves today. We'll have to wait a few days and take another X-ray to see how to proceed with the treatment, whether it will be surgical or conservative."
Nagelsmann confirmed Leon Goretzka and Corentin Tolisso are also fitness doubts for the midweek match, but Marcel Sabitzer has returned to training and may be part of the squad.
Bayern has won 66 of its 104 Bundesliga matches against Stuttgart – no other team in Bundesliga history has won so often against a single opponent.
The Bavarian giant has won 17 of its last 18 league meetings with Stuttgart, with the only exception being a 4-1 loss in May 2018.
Pellegrino Matarazzo's side has taken seven points from its last three games, however, and Nagelsmann is not anticipating an easy game in midweek.
"In terms of the table, they are not quite as good as they were last year," he said. "But it is incredibly difficult to prepare to face them.
"Their coach, who I know well, is a mathematician and that's how he approaches games. You cannot prepare for the way they will set up."
A draw for Bayern at Mercedes-Benz-Arena will ensure it sits top of the Bundesliga at the halfway point for the 15th time in the 16 seasons since the introduction of three points for a win.