Anatol, We left Taipei and headed north to Keelung, then took coast all the way past Taitung, then took 9 over mountain to Fangliao, then put bikes on train back to Taipei while we took bus to Kenting.
Renting bikes was a challenge 4 years, ago, maybe easier now. The owner of Alan's bike shop kindly set us up with 2 well adjusted cross-style bikes that worked perfectly. We are tall. The bikes were fitted with rear racks, which mated with panniers we brought from U.S. (Jandd). Might be able to buy panniers in Taipei, but we wanted to do test packing before we left U.S.
We spent 2 days in Taipei to adjust tiime, then stopped in Keelung, Suao, Hualien (2 days), ChangBin, Chr-Ben (Taitung), then 2 days in Kenting.
Very comfortable hotels in Hualien (Parkview), and Kenting (Chateau, expensive but outstanding).
The stories about nasty conditins between Suao and Huallen are true. The tunnels are very dangerous and you at least must wear reflectors to be seen and flashlights to see through the diesel haze.
A note about the entire route. Huge gravel trucks ply the entire length of the route. They are noisy, fast, and surprisingly nimble. Don't be surprised to see them everywhere. They are courteous, I chatted with one driver at a rest stop, friendly. They will avoid you, but respect their need to move their goods.
The road from Hualien to Taitung is a cyclist dream, smooth, wide shoulders, ocean, mountains and sky!
Have great time, let me know if you have other Qs.