With all do respect.
Your mother probably doesn't know the difference between slow Internet, a slow computer and/or a slow website
If i where you I'd get her to buy a new copy :p
Sad story but one where a lesson is learned
Your product key for any piece of software (be it games to 3d modeling programs to antivirus software and even Windows) is your license to use the software. (just like a cars number plate and registration number gives you the right to drive it on the road)
You could go to an Online store and buy another arma 2 cd key for pocket money. Since you still have the install cd. The advantage of buying a cd key online would mean the cd key will always be in an email. So as long as you don't delete the email you'll always have it.
P.S be sure that the site you buy your cd key from is trustworthy (Google is your friend) here are BI's official online partners http://www.arma2.com/crossroad-sprocketbanner.html
EDIT: I just read that you found it above :) good job!
Remember that you can do the above if you ever do lose a cd key
Its a good idea to copy them into a notepad file or take a picture of the back of the game manual as well (Just encase)