The Wales defender was hurt in a 16th-minute challenge with Craig Gardner as the players collided after going for the ball, with Taylor's lower leg awkwardly buckling under him as the pair fell to the ground.
Taylor was stretchered off after prolonged treatment.
"I just talked to the doctor and we are still not 100 percent but there is a fracture, at least one," Laudrup said. "We are talking about a long period of rehabilitation before he returns to the game, I can't tell you precisely how long because I don't know."
Sections of the home crowd were furious that Gardner was not punished for the challenge by referee Roger East, making his Premier League debut, though there appeared to be no malice on the part of the Sunderland player.
Taylor was in fact penalized over the incident, and Laudrup refused to cast any blame in the direction of Gardner.
"When you have injuries like that it is normally very violent, but this was not violent," he added. "It was even a free kick against us. Sometimes these injuries happen in strange ways, you don't think anything is wrong and then there is ligament damage or something like that, it is very sad.
"But you see things that are much more violent than that -- this was not violent, it was unlucky. He was very unlucky to fall in a bad way, you can't blame the Sunderland player for that."
Taylor's injury is also a blow for Wales as Taylor is certain to miss the first four games of the World Cup qualifying campaign.
Sunderland boss Martin O'Neill echoed Laudrup's sentiments and said he felt sympathy for Gardner following the home fans' angry reaction to the incident.
"I don't know why they thought that (it was a bad tackle by Gardner)," O'Neill said. "He was trying to go past and the young lad has made a foul and it's very sad he has broken his ankle. But I think the crowd were laying into him and he was surprised by that and he came over to me to say he didn't do it. He was startled and he came over to tell me. We even had the free kick, but Taylor is a very decent player and it is sad he is injured."
Gardner himself tweeted: "Hope Taylor is ok I never touched him but not a nice sight when a fellow professional goes off on a stretcher."