Providing they're in an earning Y bucket fare, (which are codes
U,E,O,Y,B,M,H,Q,V,W,T), here is a guestimate:
Durham - LHR = 300 (I'm presuming its booked into a cheap bucket)
LHR-LAX= 5456
LAX-RAR= 4670 (No cook island specified - I've taken Avarua [Rarotonga Intl], Rarotonga Island, CK as an example)
RAR=AKL= 1873
AKL-LHR (with HKG stop) = 11683
=
23982 miles.
www.milecalc.com is also good if you know your airport codes.