I stayed here
https://www.airbnb.com.sg/rooms/3187769
but to be honest I feel it is quite pricey, considering that it is far from the TRA station (not sure about HSR) but there is a public bus going there. The owner is very kind and generous, he lent us an electric bike for 2 days so that we can save on the transport fees.
Places to visit: Chimei museum, Anping old street/greenhouse/fort, chikan tower, night market (forgot the name, sorry! but there is only 1 popular one), Taijiang national park (very out of the way, only go if you have a driver. however, the green boat trail is very beautiful)
Chimei museum is a highly recommend. Tickets must be booked in advance to save you the queuing time.
To be honest though, there are more interesting places to visit in taiwan. I would comment that tainan gives me a very laid back and rustic feel, if you prefer adventures/excitement i would suggest you look elsewhere in taiwan

disclaimer: this is my highly subjective and bias opinion
