I see a number of things here. FOr a start I don't see the point of putting the damsel in there. Let the decaying material provide the ammonia for your cycle - having the damsel in there is unneccessary, and arguably cruel.
The state of a cycle has got zero to do with what your skimmer will skim out. Skimmers DO NOT remove ammonia, nitrite or ammonia - they remove organics BEFORE they mineralise to ammonia.
But the unfortunate fact is tht your skimmer is not working very well is because Redsea Berlin airlift's are in fact rubbish. I assume you are blasting air into it, and producing many small, fine bubbles. Is any foam forming? Is it simply not climbing high enough? Or is it overflowing water. You might consider a better airstone, and if you have a better airpump you could use , you might try that. But in all honesty this is not a good skimmer, and a bad choice for a beginner. I hope you got it free from an uncle or aunt.....