Assuming no delays on the train/taxi/traffic she should be good. The 1K/Biz line for check in has never been long in the 4-5times I've checked in at TPE for that flight. Security has never been a lengthy process either though once I got caught behind a tour group but it was still efficient. I beleive I usually planned on arriving around 8am.

The Singapore lounge is nothing to write home about in TPE so no reason to get there early really. It's also a bit of a hike to the UA gates.