What is the ruling and the advice of the Shari'ah regarding maintaining ties with a sister who is not visited due to her husband's ill-treatment, knowing that he has slandered the questioner?

You are not to blame for not visiting your sister as long as her husband's abuse is the impediment. Connection can be maintained through communication. Your sister should advise her husband regarding his ill-treatment of her family, and how this contradicts good companionship and the prohibition of harming people, especially relatives, and because slandering a Muslim is forbidden in Islam. So let him fear Allah, restrain his tongue from evil, treat his wife well, and honor her family.