A lot of good advise so far. IF you want my opinion, I would tell you not to do it. Yes you can save some money, but with two kids it won’t be as much as you think ,especially with only one income. You will be just as well off back home. Taiwan is not necessarily an ideal place to raise a child from my POV. I’m sure you’ve researched Taiwan a bit if you have gotten this far. Taiwan has one of the highest population densities on the planet. In the cities, like Taipei and Kaohsiung, there is little greenspace and room to run and play, living quarters are small and expensive, and the level of pollution is concerning. The cities are also, where you’ll find the better jobs and a better support system. Nonetheless, I recommend that you, as a single mother, remain close to your support system until your children are at least relatively on their own. Then you could come here and they could visit at will.
Give it the big think. If you chose to come, I would recommend Taipei if you like metropolitan lifestyle or the East if you like it more pristine and quiet. Jobs are easy to find and interviews can be set up over the internet. Looking for housing is quite easy depending on where you are moving. I recommend Tsui Mama Housing Service (http://www.tmm.org.tw). We found a decent place at a good price through it. Most evceryhtign else you can find by searching though this forum. Good luck with whatever you decide.