Kirkland, who joined the Owls in May from Wigan Athletic, was knocked to the floor in the second half of the game after Leeds United had equalised through a Michael Tonge goal.
The 31-year-old spoke to The Sun and has called for action to be taken.
He said: “I am OK now, but I was really shocked at the time.
“I was gutted at losing the goal – I had kicked the ball away and was deflated. Out of the corner of my eye I saw this Leeds fan coming towards me.
“I thought he was going to give me some abuse. Then he laid right into me. It felt like I had been hit by a baseball bat. He caught me a cracker.
“Then I am on the floor and thinking, ‘Has that really just happened?’
“I had my wife at the game, my daughter has seen it too. But this is not about me. I hope coming out and speaking means that something will be done by the authorities before something really serious happens either to a player or to fans.
“It is a sad time for football. There are people out to cause trouble. That guy has only gone to the game for that reason. It’s not about football any more, it’s terrible.
“You want families to go to games but I’m not sure you can do that any more. There is too much going on, too much abuse."As well as the assault, some Leeds fans also tore down an advertising board in the Hillsborough stadium, while various objects were thrown onto the pitch.
“I got hit by something before the second half started,” Kirkland said.
“I just want the authorities to sort this out and put some measures in place for all of us – players and fans. That is the most important thing.”