I don't have an expert knowledge of DONEs, but I'm confused with your post. I'm not clear on what your exact question is.
Do you have something ticketed already? If so, the route is basically firm, and you're just asking what's the best flight to take. This is the impression I'm getting.
From the routing you posted, I'm assuming you've to fly east. The only suggestion I can make is after AA116, catch the BA25 (I think there're 2 BA LHR-HKG flights each day). BA25 leaves at around the same time as CX250, so you get to experience T5 as well as popping into London. (Something you've touched on already.)
If your route isn't firm, from JFK, you can consider flying to HEL or Amman on your way to HKG. You can of course connect to other places if you don't mind having more flights - HKG being the most direct way to get to AKL.
I think there're flights from HKG to PER as well, but I suspect you don't want to touch those...
If you can fly west, then you can fly JFK-HKG-AKL (or ADL) all on CX. I think some of the JFK-HKG flights are also code-shared with AA as well...?