Is Mandarin important?

Not every company requires the same level! Some companies will be happy if you only have a conversational level to joke during lunch or dinner meetings with clients. Others, will not require Mandarin at all because you will be expected to deal with their international clients. Nevertheless, I would encourage you to learn Mandarin as soon as possible, one more language will always be a plus, it will always be an advantage over people that don't speak it, what do you lose by learning and polishing your Mandarin?

It depends on what kind of jobs you are looking for in Taiwanese companies. If your job is to deal with international customers, English will be your strength; but if you can communicate with your colleagues in Mandarin, I believe you can work more smoothly to get support from your colleagues. If your job is to serve the local customers, especially from Southern Taiwan, Taiwanese sometimes is more useful than Mandarin. It's not easy to learn a foreign language. As you are in Taiwan now, why not taking the opportunity to learn Mandarin or even Taiwanese? It can help you get more job opportunities and know the people here better.