The FET program is full in Taitung (i.e. they have as many people as the budget allows them to have). I know this because I’m leaving it this year and they already hired a guy to replace me.
Taitung U is possible, but be warned that the new campus is far outside the city. I worked there for a couple years, and let me tell you the head of the English Department has a big 'ol stack of resumes. I think the most effective way of getting a job there is happening upon that guy at the right moment. He’ll forget his stack of resumes if he’s looking for someone, you’re qualified, and you’re here.
Not that I’d wait around for a job at Taitung U. It’s really not that great. You can make more at some private English schools.