Growing up with a parent who is inconsistent or possibly having their own stress, mental illness is hard. None of that condones violence. Washing a child’s mouth out with soap is legally excessive corporal punishment and not allowed under the law. Your mother may know the law but she is not abiding by it.
There is so much free help out there. If there are younger children in the home, you owe it to them to reach out and get help for your mother, father, and self.
There are many organizations even in small communities that offer help. The directory or phone book will list child related services or maybe a trusting adult can talk to your mom.
As you say, your mother is a helper. There are many wounded helpers out there who need to heal themselves too. She may be open to help if she is aware of it. Calling in the authorities may jeopardize her job. Sharing a family secret is a big deal and ofcourse chilul Hashem. if you are willing to talk to an on line community, please find someone in real life that you can talk to .