How to sign into an unconfirmed Nintendo account

The game should have come with a leaflet that has the code on the back. You're not completely out of luck though, because I found this in the FAQs on the Club Nintendo website:

My Game Didn't Come With a PIN Code. How Can I Register My Game?

Answer:

If the game you purchased is eligible for registration and the card with the PIN code is missing, please contact us and provide your name, username, email address, and the game title you're inquiring about.

That question is on the North American Club Nintendo. You can select 'Other' > 'UK & Ireland' for your country, but if you're using the UK Club Nintendo website you should contact support with this form.

Unfortunately, you can't.

As Nintendo's help page about the Nintendo Network ID states:

You can only register a single Nintendo Network ID to each Nintendo 3DS system. It is not possible to register multiple Nintendo Network IDs on a single system.

A Nintendo Network ID registered on a Nintendo 3DS system cannot be registered to another Nintendo 3DS system. If you wish to transfer your Nintendo Network ID and balance to another Nintendo 3DS system, this is only possible by performing a system transfer.