I'm sorry tht you've had such a hard time with your dog!
Coughing dogs can be hard to diagnose, and the main two causes in this country are infectious bugs (e.g kennel cough, which is fairly common although more usually seen in younger animals) and heart disease, of which there are many different types.
Your dog may have heart disease. It could all start with a dodgy heart valve, as you say. The heart has to work harder to get the blood through the bad valve, causing it to enlarge. Over time blood will start to back up along all the veins because it isn't being pumped quick enough around the body - this causes fluid buildup in the tissues, especially the lungs.
Dogs with heart disease may also seem breathless or tire very quickly and can cough a lot. If your poor dog is coughing as often as you say, I'm not surprised her larynx is inflamed or sore - I know my throat hurts when I get a bad cough.
Treatments for heart disease usually involve drugs to make the heart beat faster and diuretics to try and get rid of some of the fluid buildup around the tissues. it may take a while to work out a dose rate for an individual, because every dog is different. Diagnosis involves listening to the heart as it will sound different from normal, x-rays and ultrasound scans would should a bigger heart, and lung sounds may sound raspy due to the fluid buildup. Blood tests may rule out some of the other causes of coughing or maybe show up something else. I guess these are the tests your vet has done?
If it is heart disease, one thing to remember is that it will never be cured by drugs, so it will never go away completely and your pup will always have a cough to some degree, although the drugs should reduce the severity of the disease.
Hope this helps.