Trying to wire a toggle switch and push button start

What of the original equipment was removed?
In other words, do you have an ignition switch, a starter relay still?

At a minimum you will need to crank the starter, run the fuel pump, power the fuel injectors, pulse the injectors, so the computer needs power and also power the ignition. These wiring diagrams should help with those items.
https://1drv.ms/f/s!Ald3K7ShbumWhmMox6xZjf9ZOaxL

Copy and paste that entire link.

If no ignition switch, then to crank the starter, you could grab the positive for a toggle switch from that engine starter fuse in the PDC, and the other side of the push button switch to the yellow or yellow/black wire feeding the starter relay. To bypass the ignition switch.
As far as the ignition coils, the PCM and the fuel injectors, they are appear to share the same feed from the ASD relay. It should be okay, as long as the PCM is getting power and ground.
I believe then you will need the fuse #10 in the fuse block to be hot when the key is on both run and start, or the toggle switch to be wired into or from that push button to make this hot when cranking, like when the key is on start and also this toggle switch to have yet another feed so it will be hot simulating when the key is on run. I do see this fuse #10 also gets its power from that engine starter fuse through the ignition switch, so I suppose you could use that engine starter fuse for it as well. That fuse #10 also eventually feeds the fuel pump relay, too.
That should be it as far as I can tell. Let me know if the engine will not start and what is missing if it does not, the fuel pump, spark, injectors, etc?