How do I get my Netflix account into my subscription on iTunes?

If you started your Netflix subscription outside of iTunes, then you need to Manage it and its payment method from Netflix's website directly.

Only subscriptions started directly with your Apple ID will show up on your Apple ID.

Manage Your Account - Netflix.com

If it was started on your old Apple ID account, then that's where it stays. Purchases and subscriptions are tied to the Apple ID that makes them and there is no way to transfer them from one Apple ID to another. They will not show up on a new independent account either.

You'll need to log in to that older account and manage the Netflix billing option there.

For future reference, if for some reason email addresses change, you should not create an entirely new account, but simply update the existing one to the new email address.

Change the email address you use with your Apple ID - Apple Support

Is it billed on that particular Apple ID? Do you have another Apple ID, it may be billed through instead?