Assuming everything else is plugged in properly and that you have enough of a load on the power supply to stop it burning itself out (plugging the 20/24pin connector and the 4pin ATX connector is usually enough), you can start the computer by just simulating your pressing the button.
When you press the power button on the front of a computer case, all it does is bridge a gap, provide a complete circuit and sends a signal to the motherboard to start normal bootup sequence. Without a switch, one can simulate this by using a decent size screwdriver to gently short the two connectors and start the board.