In our neck of the woods, people fall "out of" bed assuming you were under the cover asleep.
So why do we need three copies of the verb here?
Leesik and leesem, no doubt. I don't see any direct objects.
How about “ A könyv az asztalról leesik”? Does that mess up the emphasis?
That would probably be seen as placing "az asztalról" in a place of focus, requiring you to split out the preverb.
I miss the next word: and!