Ah ok, that changes things a tiny bit.
I just had a look inside my own LP style axe to understand what you need to do. I have a different wiring, 2 vol, 1 tone, 3 way toggle (same as you) and I only exchanged the PUs and put in a push pull poti as tone for splits. With the that wiring I got some more wires flying around (and I am rather tired right now, its 4:00 here ), but found what I needed to know.
Anyway, back to buisiness: As you have noticed the wire from the PUs alone will not reach the toggle, they are only designed to go through the routing and into the main cavity where all the pots reside. From there one multicore, kinda like the one from the PUs (or just multiple smaller wires) should go up to the toggle. It should still be connected to the pots if you did not remove it. If you removed it, get it back or get a new one (or multiple smaller ones).
I do not know how the original PUs were connected to the toggle exactly, but effectivly you will have to wire the red wires from the PUs to one of the cores going to the toggle each (simple extension and donīt forget the insulating tape on those solder spots, donīt them shorting out touching pots) and those two then go each to one of the lugs on the side of the toggle.
When you have the lugs facing towards the edge of the guitar (outwards), the left lug (closest to the neck) is for the neck PU wire and the right one for the bridge.
The middle two lugs of the toggle (the middle two already facing towards each other) should be connected and have a wire on them going back to the main cavitiy where it is connected to lug 3 of the volume poti (which is the poti on the left). This can be seen in you diagramm.
Finally on the toggle, oposite side of the lugs you will have to ground the toggle. That wire is connected to the inner most piece of that whole sandwhich architecture. That wire then also goes back to the main cavity and is soldered to the back of the first tone pot like the black and bare from the pots.
Cause all my talking is usually not that effective, here a pic which shows everything. Should multiply the understandabilitty of my words^^
http://www.guitarnucleus.com/gitschems/3way_LPtoggle.jpgIf anything else is unclear, go ahead and ask! I would not know half the stuff I know if I would not have asked a lot of (often stupid and simple) questions here. xD