Our roof delimna

We have had a problem with our roof for years, but haven't been able to get anyone to agree to do the job. The idiot who put the addition on our house, years before we bought it, put a flat roof on, among other stupid things, and now we are paying the price for his stupidity. The joint where the flat roof meets the sloped roof leaks all along the length of the house. Nearly everytime it rains, it leaks.

We had some estimates last year that was under $10,000, now we are getting them for $40-60,000. What gives?

I just want this project to be over with, but at a price we can afford.