Anders Hansen shot his second straight 2-under 70. He stands in a tie for third place at 4-under-par 140 with Brett Rumford, Graeme Storm, Darren Clarke, who each shot 71, and Jonathan Lomas (72).
First-round leader Trevor Immelman struggled to a 4-over 76. He fell into a tie for 14th place at 2-under-par 142.
Bjorn, who collected his eighth tour win earlier this year at the Dunlop Masters, started on the back nine and stumbled to bogeys at the 10th and 12th to slide back to even-par. He bounced back with a birdie on 13, but faltered to another bogey on No. 15 at The K Club.
The Dane chipped in for birdie on the 17th, his eighth. Bjorn, who shared second here in 2001, made it two in a row as he birdied the next.
Around the turn, Bjorn birdied the par-3 third. He got up and down for birdie from a greenside bunker at four to get to minus-4, but dropped another shot at six. Bjorn, however, birdied seven and drained an 18-footer for birdie at nine to share the halfway lead.
Donaldson, who also opened on the 10th tee, parred the first 12 holes of his round. The Welshman finally moved into red figures with a birdie on the par-5 fourth.
He came right back with a birdie on five to get to minus-5. Donaldson, who is looking for his first European Tour crown, closed with four straight pars.
Stephen Dodd aced the par-3 third en route to a 2-under 70. He shares 24th place at even-par 144.
The cut line fell at 4-over-par 148 with 74 advancing to the weekend.
American Ryder Cup captain Tom Lehman made the cut at plus-2. Meanwhile, his European counterpart Ian Woosnam, missed out on the final two rounds as he finished at 6-over-par 150 due in large part to his opening-round 78.