How can one deal with a mother who is negligent in prayer and loves backbiting, and reminding her of Allah and admonishing her is of no avail?

You must be kind to your mother and show her filial piety, for Paradise lies beneath the feet of mothers. This applies to both believing and polytheist mothers alike, as Allah Almighty has commanded. Part of showing her filial piety is to advise her gently and kindly regarding her negligence in prayer, which is the pillar of the religion. Remind her of the gravity of abandoning it, and that it is the first thing a servant will be held accountable for on the Day of Resurrection, and that it brings comfort and tranquility. You must also explain to her the prohibition of backbiting and slander, and that she should repent from all sins according to the conditions of repentance: abandoning the sin, regretting it, resolving not to return to it, and restoring rights to their owners. Seek help from tapes and sermons, and be gentle in your advice to her, and remain silent if she becomes angry.