That is not from a peck in the eye and you did right by cleaning it each day. It is from congestion in the sinus cavity and it is backing up. I am fighting the exact same thing now with (started out 8) 6 baby chicks. As soon as I noticed it I started treatment with Tetracycline and purchased an eye ointment from my Vet. I used it for 10 days and lost 2 chicks regardless. Upon realizing it was not working and almost getting ready to lose a 3rd baby, I bought a stronger antibiotic from the co-op, Aureomycin. I know it states it is for Swine Only, but after 3 days of use, I already see them getting better and the 1 that was near death is starting to move around. My Farm Vet said I am lucky that they lived and not many would care for them the way I did daily.
I washed their eyes twice a day and added the ointment and gave fresh antibiotic water daily.