22 is a good age for a cat. I would be honest with your vet and listen carefully to their advice. They would not tell you to put any animal to sleep if they did not think it was absolutely necessary. If that is what they recommend, you can stay with her (they just literally go to sleep) or, if you feel you cannot do that, then you can leave her with the vet and a nurse.
I think in your heart you already know the answer to your question.
It is probably the hardest thing that you will ever do, but try and be strong for her sake.
In time you will remember the good times, you have given her a long and happy life, she has never been hungry or known loneliness.
Take the time to grieve, don't be around people who will tell you 'its just a cat' or 'get another one'. People mean well, but they don't understand.
Not much more I can say, but listen to your vet and be prepared, also prepare your kids (don't forget they have had her ALL their life, they cannot remember a time without her).
Thinking of you.