I recommend this place. They all speak English and are very accommodating. Private place means usually much quicker and more convenient than a hospital. Cheap too, especially with insurance.
唐子俊診所/高雄身心科
No. 193號, Zhongyan Rd, Zuoying District, Kaohsiung City, 813
07 556 4028
https://goo.gl/maps/YumsNUgs8aAYYXg4A
edit: I see now that @quandary already mentioned Dr. Tang. So I guess I’m seconding him, and giving a first hand account of my experience there. All the Dr.s are professional and caring, don’t push drugs on you and work with your needs. They will also give you whatever you ask for if you have something in mind, and always check in with you in person (at no additional cost!) at refills to see how you’re feeling, how’s the medication, does anything need to be adjusted or changed, 什麼的.