Leeds said Johnson had damaged cruciate ligaments in one of his knees, while Bakke has torn a muscle in his thigh.
"Seth will be out for the long term, he will not play for the next six to nine months and I don't think we will see Eirik this season either," Eddie Gray told the club's official website. "Injuries are part of football and somehow we will have to cope without them."
Meanwhile, Stephen McPhail suffered a rib injury after arriving as a late substitute at Ewood Park, but the United boss is hoping the Irishman returns before the end of the season. Lucas Radebe will be given an extended run in the centre of the park.
Gray added: "People were a bit concerned about how we would fare in the middle, but I never had any doubts. I knew Lucas would perform and I thought he was tremendous."
The Whites travel to London to take on league leaders Arsenal on Friday.