Visa extension for Taiwan born American

Hi all, I’m new to this site and is hoping someone here can provide me with some information on how to get a Visa extension pass the initial staying period. The thing is I am an American citizen born in Taiwan to Taiwanese parents, as result of this I have the military service issue and couldn’t get a Visa for longer than 2 months. I was told that if I want to stay longer I would have to apply for an extension once I get to Taiwan, and they would either grant or deny the extension of the Visa with no specific guideline on how the application would be judged. This is a problem for me since I’m going back to Taiwan for medical treatment (knee surgeries) and need to stay in Taiwan for at least 5 months. I’m hoping to find a way to guarantee my Visa extension. If anyone here have similar experience or known people with similar issue please let me know.