I work in a large organization in the US. Because of financial issues we only have one car in my household and so my spouse (who has to be at work by 6am) have to carpool to work. For this reason my spouse drops me off around 5:30am every day for work, and my work day does not begin until around 8:00am. 

For obvious reasons I am very tired when I get to work. I have children and so going to sleep excessively early is not an option. Because of my exhaustion, I have been using the lactation room to sleep in the morning because there is a nice reclining chair there, and because no one else gets to work before 7:00am. So, as long as I'm out of there before 7:00am, everything is fine. This strategy has gone off without a hitch for over six months. 

Fast forward: last week, I overslept a bit and was found there at about 7:45am by someone seeking to use the room for its intended purpose. The person did not report me to management or anything like that, but it's clear that everyone knows what happened and my reputation has taken a serious hit because of this "scandal". People I previously had a decent rapport with are now giving me the cold shoulder, and I have heard others talking negatively about me because of this situation. I fear this may have gotten to the point where I should look for another job. But, I really love my job and I don't want to do that. 

Is there anything I can do to repair the damage I have caused? I have stopped sleeping in the lactation room but I'm not sure anyone realizes that because they aren't here that early. Should I announce to everyone that I realize that what I did was wrong and that I'm not doing it anymore. Would a gesture like ordering pizza for everyone, or bringing donuts in the morning for everyone help? I'd love any advice.