Hojlund, 20, has arrived at Old Trafford in a deal which could end up costing £72 million ($91m). Kane, who was heavily linked with United during their search for a striker, is on the verge of joining Bayern for more than £100m.
"We have chosen a striker and we are really happy with our choice," Ten Hag told a news conference on Friday.
"[Kane] is a great striker, a goal maker and apart from that he has all the conditions and abilities that you want to see in a striker. It's a miss for the Premier League, absolutely."
Kane, 30, has been a regular scorer in the Premier League for nearly a decade and reached 30 league goals last season.
Hojlund, meanwhile, is unproven at the top level, managing nine goals for Atalanta in Serie A last term.
"I don't think that I have to go into that discussion or give an opinion about that," he said.
"We are professional and the processes we do are really careful. We consider a lot of things but finally we make decisions, but not overnight. We study every decision and we are happy with the squad we have behind us now."
Hojlund is set to miss the game against Wolves because of a back injury but Ten Hag played down speculation the Denmark international could be set for a long spell on the sidelines.
"As we said previously, he had a small issue," Ten Hag said.
"He's not on the levels where our players are in this moment so we have to train him. The prognosis [is] difficult always to say. We are confident and we are positive."
Meanwhile, Ten Hag has said Harry Maguire remains available to be part of the squad to face Wolves on Monday despite West Ham manager David Moyes confirming they have had a £30m bid accepted for the England defender.
"We've had a bid accepted from Manchester United, but while he's not our player I'm not in position to discuss him," Moyes said.