The 28-year-old took a whack from the Hammers forward early in the first half of his side's eventual 3-2 defeat at Upton Park on Saturday.
Kirkland caught the full force of Cole's boot in his face when attempting to save at his feet and therefore required medical attention at half-time.
But the shot-stopper does not blame his opponent and is instead frustrated that Wigan went in at the interval 2-1 behind after Ilan scored and Radoslav Kovac converted a rebound from Mark Noble's free-kick to cancel out Jonathan Spector's own goal.
Kirkland said in The Sun: "My mouth's not too bad. It was accidental.
"I thought it might have been a penalty to be honest because I almost brought him down, but I don't know what happened after that.
"It's just a few stitches. I was disappointed with the free-kick, I thought I might have got it away.
"I tried to scoop it out. But I'm also disappointed our lads didn't follow it in and West Ham did to score."