I checked CX's website about this (dummy booking from TPE to LHR, March 2012) and you have to book at least a K fare class (Economy Core) to be able to upgrade to Business Class using miles. You also get Asia Miles from this fare class.

Note: the K fare class is not the TWD 59,000 ticket that your travel agent said you needed for upgrade, this is the cheaper TWD 42,000 ticket (according to CX booking website). Also, I don't think you can change fare class later, unless you cancel your ticket and buy a different fare class ticket.

EDIT: There also appears to be a H fare class (Economy Flex) at the cost of TWD 46,000, cheaper than the B fare class (Economy Standard) at TWD 59,000.

All costs shown exclude airport taxes, fees, fuel and insurance surcharges. If your travel agent was talking about fares including taxes, then it could be the K class fare he/she was talking about (about TWD 57,000, close enough to 59,000).@:-)

Exact costs before taxes, etc.:
Economy Special (V): 31,746
Economy Save (L): 33,846
Economy Core (K): 42,246
Economy Standard (B): 58,827
Economy Flex (H): 45,397