Provided your dog didn't eat the case or anything inside it (like the encoder wheels, shafts/bearings, or the 3" ball) then you should have everything you need.
I don't have that Happ kit but I do have a Happ 3" trackball; I'm assuming yours is a Happ also. If it's not, then disregard what follows because I've never been inside any other brand. It looks like all you should have to do is disassemble the case and pull out the two optic boards that came with the trackball and replace them with the two that came with that kit you bought.
Judging from the pic at Happ's, there's a harness that then connects to each of the new optic boards (4 pin), and then (it's hard to tell from the pic) somehow those other cables can be connected so that ultimately there's a USB or PS2 plug at the end.