Almost every hospital in Taipei has English speakers on staff. If you don't have local insurance, rates are still very cheap compared to the West. Once you've registered with a hospital, subsequent appointments can be made online.
Dr Simon Chan Clinic This private clinic of a fluent English-speaking doctor trained at UCLA is very popular with Taipei's expat population. Call for appointments.
Mackay Memorial Hospital Well-regarded private Christian hospital. Takes Taiwan's National Health Insurance (NHI).
Taiwan Adventist Hospital Well regarded for its foreigner friendliness. Takes NHI.
Taipei City Hospital Chinese Medicine Clinic Centre If you're interested in checking out traditional medicine, this hospital has English-speaking doctors. Takes NHI.