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Resistance between pins!!!!, put one probe on pin 17 and the other on pin 18, what is the resistance between those 2 pins, same thing for pins 27 and 28...
You did not answer the resistance to GND on L7220. Also if the battery is charging I'm pretty sure 3v will be present on L7220 with the battery connected and charging, check that too.
It is likely that is will power on if the battery has some charge. See if it powers on after the battery is charged a bit. Short the pads on the board to exclude a bad keyboard.