jefro - can you please tell us what your fare basis on your JFK-HKG is? Even if it is in coach, HKG and TPE are common-rated, which means from JFK both HKG and TPE are the same fare so in fact you can include a side-trip to TPE without any extra costs.

Alternatively ask CX for sidetrip options in additional to your ticket if you want to keep both of them seperate.

The CX website's fares guarantees mileage on Asiamiles only. Tickets are most likely booked in S, M or K classes.