Being on the eye is going to make this a whole lot tougher to deal with. So be careful with any of these, I don't know how many will work well/safely around the eye. Check your MSDS sheets and be prepared to do an eyewash.
I'd start with wiping it with vinegar (preferably Apple Cider Vinnegar with a live mother); this has worked very well with ringworm (a skin fungal), when that fails I go to hydrogen peroxide, also good for skin/cut disinfection.
Epsom salt wash/bath is another i use for skin infections, especially ones that start to red-line into the body; clears them right up.
Essential oils: tea tree, oregano. These will penetrate well into the body so will work well. There are others,
@Foxlily can add a few.
Tinctures (internally): garlic and others. Also cook with garlic, lots of it until it comes out your pours.
Tabasco maybe. No I'm serious, hot pepper kills a lot of stuff. I've used it for internal deworming with success.
Honey: you mentioned this. I don't know if manuka will make a big difference here or not, but it won't take much of it. There are a few ways honey is disinfecting but one is its peroxide content. You can taste this so evaluate the honey first; it should be raw, unheated. Any good honey this time of year will be crystallized. Of the surface treatments this is the least likely to bother the eyes.
I'd eat more fermented foods. I'm betting your gut bacteria are off. Also evaluate your soaps. Your skin and gut should have the natural biomes to prevent these things from taking off.
Also, check your humidity levels in the home. If its been a warm wet winter, that could be aggravating the problem and making it hard to clear up.
Probably also homeopaths that can help, but not sure off the top of my head. Belladonna is likely though.
Do you suffer from dry eyes? Tears are salty, think about it, very disinfecting. Maybe some sort of homemade eye drop that has salt? I don't know. You'll want to be careful and do some research.
