Keep her and live in Canada illegally!
Well, I hate to say it, but lots of people who have been ordered to leave Canada actually don't, and just live here without papers. The main problem would be her lack of access to free healthcare, but if you are able to provide for her health needs this may be an option.
stay here....and do not get caught under any illegal activities... ( Stay away from Police)

...you are good to go...
and if she is really old age, keep in mind that you have to bear her medical expenses if anything happens to her during that period...you might want to extend the insurance from the home country if she has one on tourist visa....
I can't believe I'm reading this "advice"!
Obviously yours is not a desirable situation to be in, but going illegal is not an option I'd consider sensible. I'd suggest instead to get in touch with an immigration lawyer to find out what your real options are at this point.
I wouldn't suggest to go against the rules and go illegal, this may turn against you sooner or later. Also, get in touch with your MP and see what they suggest you to do.