Unless your looking to teach English, finding a job in Taiwan that uses your degree seems near impossible. My expats friends and I all seem to run into this problem. Teaching isn't for us, but we want to stay in Taiwan.
I wanted to let everyone know about Expat Workforce. They post jobs from employers looking to hire expats. Most of their job post are non-teaching and often require at least two years experience. It's completely free to use, so give it a try.