I have hacked a keyboard with a header like that.  I removed the header (use a solder sucker or solder wick) and soldered the wires to where the header was.  But, you may be able to secure the wires to the header, I don't know.
Post the numbers that appear on your pcb.  (the board you removed from the keyboard)
I may have hacked the same one in the past and if so I will give you the matrix map.