What's wrong with this sentence? (French)?

Non! "Lui" is masculine and only refers to a man (could not refer to sweet Lucie). In this case, even without the apostophe, the correct sentence would be MICHEL LA INVITE A DANSER, which becomes MICHEL L'INVITE A DANSER. The example in your book LA LUI ENVOIE could also be L'ENVOIE A LUI. I know French is a pain.