I never did it before, but i just logged into gMail, and deleted notes do not go to the trash? Thats a bit alarming, so I can sympathise with you there.
So do you still have the pre El Capitain com.apple.Notes file? And is her user account still on the old machine, WITHOUT iCloud set up? Can you not just put that back.
If that doesn't work, TIME TO GET YOU HANDS DIRTY đ
Make a back up of the backup container first....
In the NEW backup, browse to ~/Library/Containers/com.apple.Notes/Data/Library/Notes/
You should see files named NotesV3.storedata or NotesV4.storedata etc. These are database files, you will need an sql database viewer to read these. If you don't have one download a free one from here, there are 2 Mac versions, download the Cipher version.
http://sqlitebrowser.org
Now, once you open the database, don't do anything. Go to File>Export>Export and export as CVS, a dialogue will pop open, highlight the ZNOTEBODY table ONLY. Once done you will have ZNOTEBODY.cvs, you can open this file up in excel or numbers, it's a simple spreadsheet.
In ZNOTEBODY.cvs you will see in column "F" your notes are there, albeit in HTML format, but thats a good thing as any formatting your wife did should be respected.
Next you need a HTML editor, either CODA or CoffeeCup will do..
https://www.panic.com/coda/
http://www.coffeecup.com/osx/web-editor/
If you've never used one before, go for CoffeeCup Web Editor, the trial is good for 30 days and it will give you a live preview of the HTML code.
All you need to do is copy the text from each note cell, them paste it into the HTML editor, it will then reformat it, removing the HTML tags. You can then copy that info back into a new note for your wife, hopefully saving everything and your marriage.
To use CoffeeCup, copy the note info one spreadsheet (note) cell at time. In CoffeeCup, go to File>New Project>Quick Project where you see a number 1 with a flashing cursor, just paste in your info, the preview window above should reflect the note as it used to be. You need to do this each time, deleted the code you just pasted, then copy a new note in from the spreadsheet. A bit tedious, but thats the best I can do for you.
Good luck, and let me know how you get on.