I suggest you drive the 3 to Sanxia, exit and take the 7 spur route down to Shimen and then connect to the 7 proper, the North Cross island Hwy. Stop at the ancient trees at Baling and then drive up to Taipingshan, a gorgeous forest reserve with some nice hot springs on the way up.
Return to the 7 and head up to Wuling Farm, and then up to Lishan and Dayuling and down through Taroko Gorge from the top (the best way down, with Chinese landscape painting views the whole way).
From Taroko take highway 9 to Liyu Lake and ride a bike or drive Hwy 14 up the Mugua River Gorge, a mini Taroko Gorge only far more wild and untamed looking.
Then drive down the rift valley. In Yuli head out to the Wallami trail. Also check out Loshan, an organic growing valley with some of the best scenery in the rift valley: think towering blue mountain ranges on both sides of you as you sit in a lush rice growing valley.
You can ride up the old South Cross Island Highway a bit also.
Then ride down to Taitung but take a side trip up to Luye, one of the nicet bits of rural Taiwan, and also the centre of parasailing. Once in Taitung take Highway 11 back up to Hualien and then the Suhua up to Ilan and taipei. Basically after Sanxia you will be in sparsely populated land with one stunning landscape after the other. And you won’t have repeated a single mile.
After Taitung you could also drive down to Kenting but make sure to take the 199. When riding back, after Fangliao take the 185 up to Sandimen. 70km of pure rural landscape as opposed to the shit you would go through heading up the coastal routes.
From Sandimen continue to Maolin and then cut over to the 20 and up the Tsengwen Reservoir way to Guanziling along the lovely “coffee road”.
From Guanziling head up to Alishan and then across to Tatajia and on to Dongpu, Sun Moon Lake, Puli and Highway 14 up to Hehuanshan where the road gets to 3140m?, putting you above the treeline.
Then once again drop down into Taroko and at the end of that head up the Suhua Highway to Ilan and Taipei.