In a statement late last month, a Nintendo UK spokesperson told Eurogamer that any consumer who may have misplaced their number or who requires a new one due to a "Wii exchange" simply has to call their local Nintendo service department and a new download code will be issued.
Thank you for wanting to put the Wii Speak Channel on both of your Wii consoles. However, since you already have the Wii Speak Channel on one of your Wii consoles, we will not be able to send out another Wii Speak Download Ticket for your new Wii. Sorry.
You only need the code to get the standalone channel; any games that use the peripheral will still work.
The UncleBob solution is to buy another WiiSpeak and use that code. They've got to be dirt cheap if you can find them, and you've made much crazier Nintendo purchases.
Didn't you install the Homebrew Channel on your new Wii? There may be a way to install it through that.
But it's something that is both free and something you already legitimately own.
Does Wii Speak work on Wii U?