Sync iOS Notes with Exchange
iOS Notes only supports syncing with iCloud or an IMAP email account, according to their 
Syncing Notes support article.  

If your Exchange server allows IMAP access, you could create another account on your iOS device that only syncs the notes for your account.  You'll have to ask your Exchange admin whether they have enabled IMAP access and what the settings are for it.

There are a handful of iOS apps which claim to sync Exchange notes, but their reviews indicate that they're unreliable, and it doesn't appear that any of them support Exchange 2010 or 2013.  As of this writing, the IMAP workaround appears to be the best option.