United made the England captain their priority striker signing at the beginning of the transfer window, only to move on to other targets over reported concerns over the cost of the deal and possible reluctance from Kane himself.
Bayern ended up agreeing a deal worth at least £100million for Kane, who turned 30 in July, as United wrapped up the signing of 20-year-old Rasmus Hojlund for £72m.
Speaking to reporters on Tuesday, Ten Hag says he is happy with the signing of Hojlund, who was injured for the first month of his time at the club, and insists United made the right decision to look elsewhere.
"I don’t think we have to repeat this debate," he said, adding: "We are happy with Rasmus, obviously first game he was injured.
"He played against Arsenal, he’s now had a start, now we have to integrate him in the team. But you can see he’s a big talent and he will contribute to our game. We’re quite confident of that."
United will face Kane’s Bayern Munich tonight in the Champions League.