The experienced Republic of Ireland international arrives on a four year contract after his contract at Burnley expired.
Hendrick played 24 Premier League games last term.
On his arrival, Bruce said: "Jeff is someone I've admired for a long time. I watched him when he was at Derby, then he went to Burnley for a lot of money and couldn't afford him, so I missed out then!
"He's an experienced international football who can play all across the midfield so we welcome him aboard. He'll be a big asset to us, on and off the pitch.
"He's a great pro, down to earth, and at the age of 28 it's a wonderful time to get somebody. It is the peak of his career - he's a really fit boy who has played 122 times in the Premier League for Burnley over the last four years, and I'm sure he'll be a great addition to the squad."