How can I quickly migrate my notes from Evernote to the new version of iOS’s Notes shipping with iOS 9? Most notes are just text or simple images with text.
The fastest way to do that would be to borrow a Mac and make a new account.
Download the free Evernote app to the Mac and sign in to iCloud and enable note syncing.
From there you can drag/drop and copy them far faster than you could on iOS.
The evernote app on iOS is intentionally designed to not let you use the inter-app sharing feature to simply move the note contents to another app.
Instead, they coded a public/private web link infrastructure that looks designed to keep your content on their servers yet allow sharing of some notes as long as you force your sharing recipient to make an account and log in to the service. Basically, your notes are in a "gated community" and notes.app isn't welcome to import that content without a fight or some effort.
Depending on how good you are at processing XML files, you might be able to export your entire notebook and process it into a form that's amenable to notes. Again, you need a PC or a Mac to accomplish that transformation as well as the initial backup.
Even better, both Notes and Evernote are scriptable on OS X, so you might be able to craft a nice tool to do the migration of the data.
OS X 10.11.4 supports Evernote import to Notes (import of the .enex file format). Imported content to OS X’s Notes can be synced to Notes on iOS devices then.
Source: AppleInsider: Apple's Notes for Mac to support Evernote file imports in OS X 10.11.4, February 9th 2016
9to5mac has a how-to for it: How-To: Migrate notes from Evernote to Apple Notes.