My experience with kids (I am now assuming that everything is EXACTLY the same as it is with kids- it seems to be true so far, even down to buying bodysuits/onesies!) is that if there is only pus, then don't worry too much, it's the redness in the white of the eye that shows conjunctivitis.
One weird tip from a health visitor was to get a little bit of the shampoo water in the infected eye when they're in the bath (making sure it's nice, gentle baby stuff, of course!), so you could try that, if you dare.
Oh, and the other tip was to get breastmilk in the eye. Perhaps dealing with babies is a bit different after all.