This may not be the case for the type of job the OP's daughter will be taking, but some companies who employ native speakers for English teaching in Japan (a common job in Japan for people from the West) do/did not cover flight costs. From personal experience some offer a fixed payment after a period of time has been spent with the company. They have to be careful as lots of people don't show up, lots of people show up and leave again almost immediately and some people turn out to be unsuitable. Again, not suggesting the OP's daughter would fall into any of these categories, but companies not paying for flights is not unheard of.

But if this company is paying, then obviously try to get as much value as possible . CX flights via HKG are a fairly long detour. From OW I've done Finnair and that's okay (read cheap rather than comfortable) from the UK. The 4 direct offerings - BA, VS, ANA and JAL are all much-of-a-muchness in Y. I've only experienced BA in premium cabins so cannot compare. Asiana via Seoul arguably offered me the best flight between London and Tokyo that I've taken as the food (even in Y) was fantastic.