Well the only things that need power on the looper are the indicator LEDs. The audio switching is completely passive. The only connection between the two is the power on / off switching through the ring to sleeve connection on the input jack.
If the LEDs don't light it would seem either you have the three wires on the power jack scrambled or the power on / off wiring is screwed up elsewhere. The LEDs won't light if - either they don't get power at all or the power polarity is reversed.
The centre pin of the power jack is minus and should go to the ring connection on the input jack and to the battery snap minus. With nothing plugged in check those connections for continuity with your meter first.
The other two connections on the power jack are its outer sleeve and its internal switch connection. With nothing plugged in these two connections should contact each other and you should see continuity between them with your meter. Having these swapped may be where your problem lies. Plug in an external power supply but DO NOT plug this in to mains power. Check that the sleeve of the power connector shows continuity to the upper ends of all the resistors (the ends not connected to the LEDs). You should be able to get your meter probe in contact with the power plug sleeve on the outside of the pedal by sliding it just a fraction out of the power socket. You can check which terminal is the sleeve on the inside of the box by connecting your meter to the sleeve on the outside and probing the terminals on the inside.
As drawn in the instructions the circuit diagram for the input jack may be a little confusing because the little triangle indicating the contact point for the tip connection is drawn upside down or at least should be moved down to the lower side of the jack. This makes it look as though the ring and the tip are supposed to connect together and break circuit when a jack is inserted. In practice with a mono jack inserted the ring contact is shorted to the sleeve providing a connection path for the minus power and battery to the ground of the circuit.
I finally had some time to look at this. You were correct. I did not realize that one of the tabs was a disconnect. I saw continuity between then and assumed they were just redundant connection points. Once I determined which was which and got everything rewired correctly it worked great.
Just for grins this is a pick of the pedals I have built myself so farhttp://members.toast.net/dwyss/pics/pedals1.jpg