Avoid a triple-leaf construction. If you build a 'room-in-a-room', tear off the existing drywall first. It's been mathematically proven that for a given mass the most efficient soundproofing construction is double leaf. Ideally, you want 2 massive layers, independent of each other, with a large airspace and full of soft, absorbtive insulation. Don't forget sealing and use 'airtight' electrical boxes if possible.
If you go with isolation clips you'll still want to tear off the existing drywall. If you don't, the wall will suffer - as much as 20 pts STC.