My apologies - the exceptions for second entry to Africa via London are Ghana and Nigeria (not Kenya). So Nairobi isn't possible. You could replace NBO with Lagos (LOS) but you still have a problem in that your itinerary enters Europe three times - the max allowed even with the ACC/LOS exception is twice.

Here's a variation which I think would work:

ord-lax-sju-jfk-phx-jfk-eze-jfk-syd-nrt-hkg-cmb-hkg-jnb-lhr-los-lhr-dxb-lhr-lax-ord
(75717 miles)

But you could achieve your objective of EXP at lower cost by reducing the number of continent; here's an AONE4 that would achieve it:

ord-lax-sju-jfk-phx-jfk-hkg-khi-hkg-khi-hkg-jnb-nbo-jnb-lhr-acc-lhr-dxb-lhr-lax-ord
(67147 miles)