You may need extra wire for that, the wiring that comes with the ZD isn’t very long. the splicing method I used was to join the signal wires for each button at the existing terminal strip, which required new .110 disconnects. If you take that route, buy more than you need - you’ll probably end up having to redo some of them. Also, you’re going to have to solder the connection for the home button, due to the way it’s connected (with the Cthulu, you can make start + back function as the home button)
Just thought of something else that will save you work and clutter. The ZD’s included wires have a signal and a ground wire for each individual button. Since you’re tying into a TE, which uses a terminal strip for signal and ground, you could join 1 ground wire from the ZD to a single ground QD in the TE and cut the rest of the ZD’s button ground wires off. Use a multimeter to be sure which wires are ground before cutting them (connect the wires to the ZD, set your multimeter to ohms, then probe two wires of the same color. If you get continuity, they’re ground. Should be the black wires, but again, you want to be sure).
As for connecting the stick itself, given the short length of the harness included with the ZD, I suggest separating the wires from your existing JLF harness and splicing in with small wire nuts. Like I told you earlier, you’re probably going to have to do some directional swapping for the ZD’s harness (fortunately the ground wire is correct), so test it and get them straight before closing your case back up. Wire nut splices are easy to swap around, or you could remove the connector from the ZD and swap pins around.