Why are we G-d's "servants/ slaves"? He doesn't need any servants/slaves. How do we understand, "Go and do mitzvos, because we are his slaves!" (Ana avda dikudsha brich hu.)
The logic of "why we are G-d's slaves" seems simple:
It is axiomatically accepted that if you make an item it is automatically yours - you're the rightful owner of it. G-d created the matter - so the world is His, He created Man from dirt, so Man is his. Therefore legally we all belong to G-d.
Judaism holds that all the providence comes from G-d, He feeds us and provides shelter, etc. If so, we owe Him.
To "pay" Him back, according to His will expressed in the Torah, we do the Mitzvos as commanded and that levels it off.
Second, does He need our work? In our eyes yes, as we need the providence. In His view - we can't possibly judge.
Third - where do you see the word Mitzvos in אנא עבדא דקב"ה?