What is the religious ruling on dealing with an aunt who slanders her relatives and seeks to divide them, and verbally abuses her children and neglects them? And what is the ruling on lying to her to maintain kinship ties without her knowledge?

Your aunt's action is impermissible, as it causes a severance of kinship ties, which is one of the major sins and a cause of Allah's wrath. Therefore, she should be advised gently and reminded of Allah and the dire consequences of her action. As for you, you have done well by not responding to her and by your eagerness to maintain kinship ties. Doing so without her knowledge is preferable to avoid her anger. Instead of outright lying to her, use circumlocution (Ma'areedh) to avert harm.