Cat with Runny Eyes

I have a kitten (adopted from a shelter at 3-4 months) for several months now (she's 7-8 months now). When we got her home she had a discharge from her eyes, the vet put her on lysine, and it didn't do anything to help. She mostly gets the discharge when she's eating (head down), and in the morning I clean her eyes if there is accumulation. I do not wish to give her antibiotics and was wondering if there is anything natural that I may do to help her out?

What does the discharge look like? What is the colour and consistency of the discharge? Is there any associated odour? Does she scratch at the area? Is there any discharge from her nose or her ears? How much water does she consume? What kind of food is she eating? Cats who display a thick, yellow-green or gray discharge from the eyes may have some issues with digestion and liver function, particularly when the discharge is most noticeable first thing in the morning and after meals. If this is the case, you may want to consider preparing her meals yourself. Please see the work of Richard Pitcairn, "Natural Health for Dogs and Cats", "The Complete Herbal Handbook for the Dog and Cat" by Juliette de Bairacli Levy and "The Barf Diet" by Ian Billinghurst for further information and ideas for recipes.
A good, homemade diet can go along way to strengthening the body, improving digestion and maintaining good health. In addition, you may want to try a liver tonic like Dandelion root (Taraxacum officinale) in her food to support the function of her liver. If you can, try to find a glycerine-based tincture of Dandelion root; this should be available at your local health food shop or through a professional herbalist in your area. The recommended daily amount of Dandelion root tincture is: 0.5-1.5 ml per 10 kg (20 lb), divided into three equal doses per day (Wynn & Fougere, 2007) and can be added to her food.