Yes this is the most likeliest cause of the problem. That wire is responsible for supplying power to the entire top layer of LEDs. So unless you have every LED on the top layer the wrong way around, which is unlikely as it looks like you've got them all the correct way on the lower layers, then that's the probably where the problem is. Just give it a check for a bad connection or bad solder joint. Also check that the pin circled in red in the image below is soldered correctly:
For the 5mm LEDs on the corners, these are all independently powered from the 5V supply and should light the moment power is connected to the cube. For all of them to not work there would have to be 4 separate faults which is unlikely so my suspicion is you may have soldered the LEDs the wrong way around. There is a small flat side at the base the LED but its difficult to see. If you can find which side of the LED it is on it should face to the outside of the cube for each LED.