Does your friend speak Mandarin? If so, all the major hospitals in Taipei have one or two psychotherapists and definitely psychiatrists to help with intervention. (It might be the same in Taichung.) If she only speaks English, I have read somewhere on this site that there is a community counseling center in Tianmu. If your friend can read in Chinese, there are tons of chinese websites in TW to give you local service provider information. If not, then at least there are a lot of websites in the US that can help her to learn more about depression if she is interested, and there are also various support forums for various disorders. Sometimes a little psychoeducation can go a long way and make a person feel less alone.
As far as I know, (and I did this little research when I went back last december,) with hospitals in Taipei, you walk in for an appointment and you wait for the psychiatrist to see you. (You must see the psychiatrist first and they will refer you to a psychotherapist/counselor if they assess you to be in need of that service.) My research stopped there since no one I knew wanted to actually go in for a visit, and I felt bad to fake it myself just to see what happens next. One great thing is that if you were to see a psychotherapist in the hospitals, you only pay a minimal fee and the rest is picked up by insurance. However, I don't know about the quality of therapy conducted in Taiwan. The modalities seem to be very very similar. I just don't know how it's applied.
The most important thing is to prepare for your friend that she is the one in control of what kind of treatment she receives. A lot of times clients get very discouraged when people who are supposed to help them fail to do so and they give up seeking help (but of course). If she doesn't get what she needs with the first try or run into challenges with the admin side, then it's important to keep searching for the right person to help her. Someone will at some point, and it usually helps a lot for someone to speak to a counselor/therapist if they were that low. It's not supposed to be this hard to access mental health care, but Taiwan is just starting to develop its community mental health care, so. I don't know anyone who's gone for therapy or counseling in Taiwan, so maybe it's easier than what I have learned through websites and from talking to people.
Does she have any local friends? Anyone that she can check in everyday with? Sometimes that helps to regulate a person to get through the day without carrying out the suicidal thoughts.
Hope this helps. Hope your friend gets what she needs and makes it through. A lot of people do die from suicide. It's the eighth cause of death here in US. It's great that you are trying to help her.