Nagini is a horcrux and can't be killed with a wand! The only reason the Sword of Gryffindor works is because it absorbed the traits of basilisk venom. Basilisk venom is one of the only things that can destroy a horcrux!
Professors Snape, Dumbledore and Slughorn in the back ground is gold! They may be extremely powerful, wise and all knowing- wizards and yet teenage girl drama fascinates Dumbledore, baffles Snape and terrifies Slughorn.