Not sure what the problem can be, I had to do this last month and I encountered no problems.
The PS4 wanted me to Verify my Account and sent the code to my old email address.
When I went to sign into this Account, it flagged up as gone like in your case.
I simply remade the Email Address again, logged back into PSN and resent the code.
It arrived moments later.
Your email has to be exactly how it was before it was deleted.