Taking a taxi would be really expensive.
A better option would be a bus, there are buses that run every day between Taichung and Sun Moon Lake.
Bus From Taichung
Nan-Tou Bus (南投客運) Running daily from Taichung Kan-Cheng Station (干城站) at Shuangshi Road (雙十路, near the railway station), High Speed Rail Taichung Station, Puli to Sun Moon Lake for NT$200 one way. For timetable and other details, please contact +886 49 2984031 (Puli)
Renyou Bus (仁友客運) has less frequent departures from Liouchuan East Road (柳川東路) near the railway station for NT$200 one-way and NT$350 for a return ticket.
GuoGuang busses from Taipei stop over in Chaoma station in Taichung.
This is the bus and train schedule for Sun Moon Lake from Taichung and Taipei.
I will try to look for more information and see about the cost of a taxi, even though I think it would be very expensive.