Getting to the airport about 8:45 should be OK. Security is fast there. Immigration should also be OK. Definitely do Online Checkin. Since that flight tends to board early and depart the gate early, be sure to get to the gate at least 15 minutes before departure time.
The HSR is pretty prompt but once a while things go bad. Then the last few miles are the most uncertain.
It will cost more (1 extra HSR fare and hotel) but maybe you can go up with her the night before and take the HSR back to TNN in the morning?