Is it permissible to give zakat to the wife's mother, knowing that she has a well-off husband and working children who are content with their salaries?

There is no objection to gifting or assisting the wife's mother from money other than zakat. As for giving her zakat, Allah has specified the recipients of zakat, and the wife's mother whose husband is alive and has a salary, and who has children with salaries, they must spend on her if she is poor. In that case, she is not eligible to receive zakat.