Is your Apple ID an e-mail address (new IDs need to be e-mail addresses)?
If so, enter that e-mail and password in the app on you other iOS devices.
If that doesn't work, try logging into the Netflix website with your Apple ID e-mail address.
And if all of that doesn't work - call the fine folks at Netflix. They're open 24 hours.
You should be able to use your ID e-mail address on any Netflix device.