Where are the power-on jumpers on a mid-2013 MacBook Air logic board?
They are located just above the airport card.  A small flat head or T7/T8 bit should work for bridging them.
You can see them in step 17 of iFixits Logic Board replacement guide:

Power Pads highlighted with a red square along the top edge of the Mac, picture courtesy of iFixit
Something to keep in mind is that a damaged or faulty keyboard could prevent a Mac from starting if the power button is always registering a press. If I were troubleshooting this I would first try to short it using the power pads. If that didn't work after a couple of attempts (the flat ones are more difficult to short) I would disconnect the keyboard's flex connection (to the right of the battery connection) and attempt it again.
Obviously, always use caution when working inside of a Mac that has power supplied to it.
Good luck!