I can't switch text messaging back from my email address to my phone number. iPhone 6 iOS 9.2
The loading wheel indicates that the servers that handle messaging on Apple's end are still busy processing your request.

It's kind of a busy day and you might have done everything correct - but if the servers are overloaded, you won't be able to use the activation / iMessage.

Some things to check:


http://apple.com/support/systemstatus
https://support.apple.com/en-us/HT201422


The second link is a great checklist - if you've done everything listed, it might be time to contact your cellular carrier and have them verify you can send/receive SMS. Once that's done, Apple's support is the next step to have them look into your account specifically if there isn't a general slowdown or outage. It does say that 24 hours can be the time needed for an activation to be processed. I presume they have engineers that look at all the failed activations and process or clear them manually or programatically - but I can't say that for certain.