Much as I hate to say it, you may wish to consider putting drops into the good eye at the same time, since there's a risk that the conjunctivitis could jump from the other eye, that's what happened with Zev.
I find it easier (!) to do, with Zev inverted, even though she hates that position. With her upside down, you can just place the drop into the corner of the eye, and it will run over the entire eye.
I drop the towel over her, and then pick her up, turning her over, wrapping the towel around her, and over her paws (it doesn't matter too much if her rear paws are sticking out a bit at this point. She'll tend to fight, but her claws then get caught in the towel. It's important to get it fairly tight around the neck, so she can't get her front paws out. You may need to redo it, if they escape.
Then with her head on the left, I hold her head with my left hand, over her ears, and down to her jaw (partly from using those fingers to help force her mouth open, to give pills). I pull her head back slightly to help make her open her eyes, you can use your fingers or thumb to slightly pull the skin down a little, to aid that. Pressure through the right arm at the same time holds the towel around her, and holds her on my lap. I administer the drops using my right hand, which is naturally in roughly the right position.
(I'm right handed, if you're left handed you probably want to do it the other way around).
That may send a bit involved, and you need to be firm, but it lets me put drops in the eye of a cat who dislikes being picked up, and hates being turned upside down, unlike Talisker, who was happy to be held like that.