As others have guessed, this is probably a 4*5 matrix keyboard. This guess is supported by the labels of the two ribbon connection areas, R1 through R4 (rows 1-4) and C1 through C5 (columns 1-5). My guess is the LEDs are just backlight, also supported by the labels where there is only a LED+ and a GND (matrix keyboard wiring doesn't require GND) so the LEDs are likely not individually addressable.
I agree that the keypad library would work. In order to figure out the row/column assignments for the buttons instead of trying to follow traces just use the library and output to the serial monitor. Modifying the example on the keypad library (see Erdin's reply for the link):
#include <Keypad.h>
const byte ROWS = 4; //four rows
const byte COLS = 5; //five columns
char keys[ROWS][COLS] = {
{'R1C1','R1C2','R1C3','R1C4','R1C5'},
{'R2C1','R2C2','R2C3','R2C4','R2C5'},
{'R3C1','R3C2','R3C3','R3C4','R3C5'},
{'R4C1','R4C2','R4C3','R4C4','R4C5'},
};
byte rowPins[ROWS] = {5, 4, 3, 2}; //connect to the row pinouts of the keypad
byte colPins[COLS] = {10, 9, 8, 7, 6}; //connect to the column pinouts of the keypad
Keypad keypad = Keypad( makeKeymap(keys), rowPins, colPins, ROWS, COLS );
void setup(){
Serial.begin(9600);
}
void loop(){
char key = keypad.getKey();
if (key != NO_KEY){
Serial.println(key);
}
}
This should hopefully tell you on the serial monitor which row and column a key is when you press it. (I haven't tested this code, hopefully it works with multi-character strings in the keymap array "keys". If not, I suppose you could use 0-9 and A-J in the keymap array.) Once you have mapped this out (possibly marking on the board itself next to the contact area with row and column), you can change the constants in the keymap array to something that actually makes sense for your end application.
I know you don't care about the LEDs, but in case that changes in the future... I don't see any resistors on the board, so if you want to use them at some point, it would probably be safe to assume that you need to provide a current limiting resistor off-board. I also can't tell how the LEDs are wired (series, parallel, or some weird series/parallel combination). I do, however, see the flat on the LEDs and their respective silkscreen shapes indicating the cathode (minus side) of each LED. Use of an ohm-meter in continuity mode along with a dose of mental logic might help solve this black-box problem...