Okay, lesson time!
The big box is a transformer. The power coming from your wall is ~110v alternating current, but the solidoodle uses 12v direct current. The transformer drops the voltage and rectifies the AC to DC.
Coming in to your board you have 2 wires. One is +12VDC and the other is 0VDC. If you confuse them, your board will go pop (unless it has a couple diodes for protection, which it most likely does not). Essentially, you have to get that 12V to the board.
That power adapter you linked from sparkfun is the right idea, but it doesn't have enough amperage to supply a 3d printer. What I would probably do is use the existing power supply and add an inline barrel connector. The jack you linked would work (one of the pins is an interlock and can be disregarded for our purposes) but it doesn't have the right pins to fit in your board. I will take a look and see what I can find for you.
SD3 w/ RUMBA (8825s), merlin hot end (in pieces). Ender3 w/ silent board for PLA printing. Ender5 w/ silent board, e3d v6, new z lead screw, and glass bed. DiY Kossel w/ smoothieboard (in pieces). Vellman Vertex (in pieces)
Shapeoko2 router in process of being converted to laser engraver (in pieces)
Multicam 5000 series CNC Router w/ 11HP spindle, 5x10' table, and auto tool changer (in pieces)