Earlier in The King's Speech it is shown how he can speak effortlessly while listening to music through headphones. So why didn't George VI simply deliver his address while listening to music on headphones?
Probably because Lionel was trying to cure him once and for all rather than becoming dependant on the headphones, using them as a crutch. As an example, the King wouldn't have been able to use headphones at public speaking engagements. It's not like he could give up talking in public, so a more permanent cure would have been needed.
All the headphones did was give the King the proof that he didn't have an untreatable problem.