Hi there - I'm Dr. Sara. I'm so sorry to hear of the troubles that you're having with Lexi - it sounds like she is very sick :(
When I see young pets with kidney failure (diagnosed by looking at blood work and urinalysis results), there are a couple of diseases I suspect. Acute kidney failure can be caused by infections - depending on the incidence in your area, I would consider a diseases called leptospirosis. Leptospirosis is a bacterial disease harbored in the wildlife population and spread by contact with infected urine. We see a good bit of leptospirosis around here - it causes severe acute kidney failure. Another disease that we see frequently here is lyme disease. Lyme disease is transmitted by ticks and when it causes kidney failure it is almost always fatal within weeks to months. These diseases may or may not be present in your geographic area - ask your vet.
Another possibility for kidney disease in a young dog is congenital kidney disease. This is where the kidneys never formed correctly. As the pet grows, they outgrow their abnormal kidneys capacity to function and they begin to go into kidney failure.
When I diagnose kidney disease in a young dog I always recommend further diagnostics to identify what's causing it because some diseases like leptospirosis can be treatable and the pet can make a full recovery. I usually do an abdominal ultrasound to evaluate the structure of the kidneys, a urine culture, and send tests to the lab for leptospirosis. Depending on how common lyme is in your area, that test may be helpful as well.
Unfortunately it sounds like Lexi is a very sick girl. Managing kidney failure includes using medications to control their nausea and high stomach acid levels as well as fluid therapy. Sometimes they also need potassium supplements or phosphate binders (as dictated by their blood work) or medications to control high blood pressure that can result from the kidney disease. It's difficult for me to say which things might be helpful in her specific case without seeing her and her lab work myself, but the above things are some treatments to maybe discuss with your vet.
The cough and eye discharge doesn't really seem to fit for me. The only thing I can consider is that maybe she is so ill from her kidney disease that she has become susceptible to other infections. If I'm treating a cough that isn't responding to my antibiotics, I move on to take some chest X-rays next.
Ultimately in many cases kidney disease is eventually fatal - we are just managing them the best we can for as long as we can. Unless its a curable type of disease, there comes a point where no more medical intervention is going to help, and we have to discuss the difficult decision for euthanasia.
I sure hope that this information gives you some talking points for your vet. Please let me know if I can answer any other specific questions for you.