All of the switches and LEDs from the buttons will share common ground. In order to do this, we'll need to create several short wires.
Use silicone ribbon wire to create 32 short wires that are approximately 70mm(2.76in) in length.
Use wire strippers to remove a bit of insulation from both ends of each wire. Tin the exposed wire with a bit of solder.
Apply a bit of solder to all of the pins on each button. This will make attaching the wires to the pins easier.
First Ground Wires
Attach two wires to the ground pin of the first switch. Reference the markings on the buttons (if they have them).
The pins inside the gray box are the pins for the switch. The pins outside the gray box are the pins for the LEDs.
Each ground connect will need to jump to the next button in the arrangement. Using tweezers can help hold two wires in place while soldering.
Starting with the first button from the top right, proceed to wire from right to left to complete the first row of four buttons.
On the four button, jump to the next row with button #8 and proceed to jump from left to right.
On the fifth button, jump to the next row with button #9 and proceed to share ground from right to left.
On the twelfth button, jump to the last row with button #16. Proceed to share ground from left to right.
Button Switches Shared Ground
Take a moment to double check the wiring and ensure the all of the solder joints are solid. Lightly tug on the wires to check if they're secured.
LED and Switch Shared Ground
On button #13 in last row, connect the second ground wire to the ground pin on the LED. Proceed to share ground to all of the button LEDs.
Button LED Ground Wiring
Complete the last row, Button #13 to #16 and connect each going right to left.
Proceed to connect Button #16 to Button #12 going from left to right.
Proceed to connect Button #9 to Button #5 going from right to left.
Proceed to connect Button #8 to Button #4 going from left to right.
The last ground wire connects Button #2 to Button #1.
Once complete, take a moment to review all of the wires.