Am I right in thinking that for each plane you have a sepreate cathode connection and that all the cathodes are not connected together on each plane? Otherwise it would never work would it.
Have you done any tests to see how these LEDs look when they are on, if you need diffusers or what current you need to run them at.
Once you know the current for each LED you can then work out how much current to sink with the cathode sinks, you might need a FET here.
You then can look into how to provide that current at the anodes.