There are only 2 options, really.
1. Private cab / taxi
I have no idea how it's going to cost. But it's probably pretty expensive.
2. CingJing bus (E-Go)
Duration: 4 hours, once a day
Cost:
One-way: NT780 (weekday) NT880 (Holiday)
Roundtrip: NT1260 (2 weekdays) NT1360 (1 weekday + 1 holiday) NT1480 (2 holdays)
Refer to their website for more details.
http://www.cingjingbus.tw/ratesaschedule.html
Not going to Taipei? You might find this useful.
CingJing - Sun Moon Lake
