Your problem is the specific dates. When I used KVS to find award space to use my old BA EC miles, before the switch to Avios, I could only find space in J on SFO, LAX or YVR to and from HKG on CX for about 3 days in any given month. (YVR had nothing)

Fortunately, we were looking to go to Indonesia (into CGK, out of DPS) for about two weeks out of a three-month span of time, Q2 2012, so we were eventually able to get tickets ex-LAX on the newest (Cirrus) business class 77D.

So, if you have to use miles for one of the tickets I would check every gateway city and be as flexible as you can. But you might have to earn rather than burn on this trip. It might be worth paying for Diamond KVS access too - it's certainly easier to find space that way, but, be sure to read all of the details, as CX does not release award availability to all partners evenly.