Yes, you can surely get a teaching job in a high school in Taiwan with a non Education focussed bachelor degree. Having the TESL certificate will also boost your chances.
According to the
Department of Education in Taiwan,
1. Colleges: The city has two colleges, namely Taipei Municipal Teachers College and Taipei Municipal Physical Education College
2. High Schools: The city has 26 public and 22 private high schools, as well as one under construction. Seven vocational high schools in the city offer general high school education program.
3. Vocational High Schools: The city has 7 public and 11 private vocational high schools. Eleven private high schools in the city offer vocational education program.
4. Junior High Schools: The city has 57 public and 4 private junior high schools, with 21 high schools offering junior high program.
5. Elementary Schools: The city has 139 public and 10 private elementary schools, and 4 are under construction.
6. Supplementary Schools: The city has 31 public and 20 private supplementary schools.
7. Kindergartens: The city has 131 public and 251 private kindergartens.