Jeffers was shown a straight red card in the 37th minute for an off-the-ball clash where he appeared to headbutt Vale captain Tommy Fraser.
The Owls went on to lose the game after second-half goals from Kris Taylor and Rob Taylor and Jeffers was ashamed of his actions.
"I did not sleep well and I wanted to make an apology," Jeffers told Sky Sports News. "There were a few incidents before and then bit of a coming together between us and a clash of heads.
"The manager did not say to much to me, he was just disappointed to have lost the game. I've let the club down, the players, the manager and the fans and I just wanted to say sorry."
Asked if he would contact Fraser to apologise personally, Jeffers added: "I'm not sure yet. I expect I'll try and get the lad's number but I think he will understand why it happened."