1k ohm resistor seems a bit much, the 270 ohm on your planes with no resistors on your columns sounds closer to what you probably want. I dont really see the need for the ULN 2003a Darlington arrays in your setup there, You should be able to run 27 LEDs without all those parts, heck I have a 64 LED cube that ran on an uno and 4 resistors (it works better with 16 resistors on the columns.)
Just a guess, you could probably put 150 ohm resistors on the columns( 9 ), and do away with the rest of the parts, and have a fine working cube.
The exact resistor value depends on how much current you need to fully illuminate but not damage the LED. Typically with 5mm LEDs they are often 20ma, and the forward voltage is about 2volts, You are sending it 5v, which could damage the LED, so you should resist it down to about 2v (or whatever the datasheet for that specific LED says).
You may want to do a little testing with different resistors, or a small pot to find the brightness level/voltage/resistance you like, and keep it within safe levels.
Once you get your cube working, adding complications like transistors, darington arrays and such will be a lot easier (although, with a small cube like that, using all those parts are not really necessary, but is NOT a bad way to learn how to use them, on a smaller scale.)