We're only talking road bikes here, not MTBs or flat bar hybrids.
There's a FUJI Sportif on there that can get the job done.
Thanks for sharing @gosfordmonkey
Here's a update for 2017:
If you're visiting Taipei and in need of a rental (TOP two locations speak very good English):
Auranden Bikes in Tian Mu https://www.facebook.com/aurandenbikesHigh end. Price unknown.
Specialized in Da Zhi https://www.facebook.com/specializedtaipeiHigh end. 2000NT for any model and size for a day. You can use the rental fee to buy merch in the store.
Giant near Da Dao Cheng Wharf https://www.facebook.com/profile.php?id=100001578468797&hc_location=streamMid-low end. 1000NT for any model and size for a day.
Bufalo near Da Dao Cheng Wharf https://www.facebook.com/bufalobike/Mid-low end. 1000NT for any model and size for a day.