Hsiadogah posted a useful link the other day. It is most useful if you can read at least a little Chinese.
The page you want is; thb.gov.tw/damage/nowa.asp
From there you click on the dot that represents the county you want to find out about. It seems to be that if the dot is red it means that there are some major road closures due to damage in that county. Perhaps Hsiadogah could confirm.
Once you’ve clicked through on one of the red dots, you’ll see some information in Chinese and there may be one or more links to .jpg map files showing damaged areas.
It appears that your intended route is OK at the moment, but I hope someone with better Chinese skills than I can confirm this.