N.S.
Doctors annoy me sometimes!
Her tonsils are large because she has an allergy! Removing them would be solving the symptom, not the problem.
Sometimes allergy testing turns out negative, or kids can be allergic to things that aren't tested for. The best thing you can do is do an elimination diet and see if you can discover what it is for yourself. Doctors just want to prescribe meds and do surgeries. If she's allergic to a food (like dairy) then removing it from her diet will ease the symptoms. Give at least 4 days of not having that food to see if there is an improvement (it takes 4 days for the food to leave her system) and keep a log.
Also, try putting a good quality air cleaner in her room. I used to wake up with a runny nose all my life. Finally, at the age of 30 it was suggested I get an air cleaner and put casings over my mattress. That helped! The casings kept the dust mites out (nothing you can do about dust mites) and the air cleaner kept the air free of allergens. I suggest a Honeywell air cleaner because you don't have to replace the filters every few months. Just vaccuum it off. You can get a Honeywell at Target or Walmart. I'm sure your home is sanitized, but there are some allergens you can do nothing about.