According to Feiren,
"Well, you can, but it's complete madness.
If he's is good shape the OP should consider riding out to Yilan on the Northern Cross (7乙) from Sanxia through Fusing (7) and then turning right on the Lishan branch (7甲) of the Central Cross at Cilan. At Lishan, turn onto 8 and follow it all the way down to Taroko. You have now bypassed Suao-Hualian highway and ridden through some of the most beautiful parts of Taiwan. Reward yourself with a fresh from the orchard snow pear in Lishan--expensive but well worth it. You'll have earned it.
From Taroko, take H9 to Hualian. When you come out of Hualian, go dwon to the coast where 11 begins. But don't take 11 just yet...the first 25km are boring and often very hot. Instead, at the Highway 11 junction, take 193 down the east side of the Rift Valley. The best way to get over to the coast is a little road that will take you over to Shuilian just before the climb up to Baci. This tertiary road can be tough to find (no English signs), so if you can't find it, cross over to 11 at Guangfu on 11甲. From there it's a straight shot down to Taidong.
It should take two days to get to Lishan. One long day from Lishan down to the Shitiping fishing harbor at around 63km on 11 about 10km south of Fongbin. One more day to get down to Taidong for a total of 4. You could break things up by spending an extra night in Hualian.
Another alternative would be to catch a bus in Luodong for the ride across the Suao-Hualien highway. They will fit in the bottom compartment if you take the front wheel off. You could also take 106 out to the coast from Taipei and then head down to Yilan on 2甲 if you do not want to do the Northern Cross. But that would be a mistake because the second half from Baling to Cilan is incredibly beautiful.
Based on his route, I have come out with the route on Google maps. I just don't get it when he mentions "take 193 down the east side of the Rift Valley. The best way to get over to the coast is a little road that will take you over to Shuilian just before the climb up to Baci."
Excited... since this plan is firming up well, I think Imma buy my tickets! But to Taipei or Kaohsiung? Advice desired!