There's only a few places in Taipei that rent out road bikes.
At the corner of Minsheng and Huan He, there's 2 stores that rent out road bikes. The Buffalo Store and the Giant store both rent out road bikes.
You can also head all the way up to the Giant in Bali to rent road bikes there too.
Rates are 1000NT a day from Buffalo. I assume if you plan on renting for that many days, you can try to strike a deal with them.
Not sure aboutr the other stores.