Which country are you flying from ?

United States. Los Angeles.

I am sure others from the West Coast will give you some ideas.
A few pointers though, if you are going to be traveling at your own expense try and stick with an airline where you can build up mileage in that area, Cathay Pacific would seem a good option.

I don't know the tax side on claiming expenses but if the bottom line is that it comes from yuor pocket you might think about using say Eva Air Premium Economy ( just search for Eva Air on this board and you will find lots of information ) as a way of getting reasonable comfort without the level of Business Class to get you via Taipei to HKG, from HKG up to Shanghai and down to Jakarta maybe check what a local HK agent can do for you, tiglion.com have been mentioned, perhaps have a look at their site and see what they offer HK to Shanghai and down to Jakarta, then weight that up against what thru tickets would cost from LAX.
I am 'guessing' that HK is fairly competative from LAX.

Another idea, if you are going to travel several times a year is to check flight 'ex' HK back to LAX and if the fares are cheaper what you can do if buy a 1 year return LAX-HKG and keep it 'open', once in HK but locally a discounted return HK-LAX, use that back home, use it next time to get 'to' HKG and keep rollong on that way. So only once per year are you using a fare bought in LAX, otherwise flights would be bought in HKG.
This is all if it works out that tickets are much cheaper ex-HKG, it will be useful testing this theory out!