Hi!
If your visa is expendable, you can stay up to 6 month. All you need to do find a legitimate reason for staying longer and then go to the headquarters for visa extension. You'll need to go there before your visa has expired! Don't be late!
Purposes for staying in Taiwan can be attending Chinese classes, or other studies recognized by the government.
One of them is Association of teaching Chinese formerly known as Pioneer language Institute. It's on Hoping E road Nr 200 6 floor near Shida university.
You'll have to register there first, get some papers from them and then apply your visa extension.
Good luck