I did it about 5 years ago.
The thing is, riding 'around the island' is pretty cool, but the best roads go over the mountains and you can skip much of the W Coast.
So I came up with this route, which I was very pleased with.
forumosa.com/taiwan/viewtopi ... it=scooter
In a nutshell:
Go over the Northern Cross Island to Yilan, down to Hualian. Take the inland route from Hualian to Taidong. Over the Southern Cross Island to Gaoxiong County down to Kending. Back up the East from Kending to Taidong, thent he coastal from Taidong to Hualian. Up through Taroko and over the new central to Nantou. Down to Jiayi and up to Alishan then via Yushan to loop back to Nantou. Back over Hehuanshan to Lishan and down the 7甲 to Yilan then around the coast back to Taipei.
But I guess the Southern isn't open anymore.
I did that route in 5 days, but most of that was close to ten hours a day. 10 days would be cool.
Don't miss the (new) Central up Taroko and over Hehuanshan. You can stay at Qingjing. Muchaman hates it, but it's great.
If you can only pick one, the inland (Eastern Riff Valley) Hualian-Taidong road is nicer, imo.
Alishan to Yushan to Sun Moon Lake is a pretty cool road that you'd easily miss on a normal 'round the island'.
Watch your throttle hand. Rest it and stretch it. If your throttle is tight, get your mechanic to loosen it. I got some pretty bad strain that lasted for weeks after the trip.
Edit: Oh yeah, reading what Lo Bo To just said. Get a decent check-up on your bike before hand, and maybe another halfway through. New tires, checked brakes, oil changes etc. I did 2000KM, which is more than I usually do in 2 months and I ride every day.