Christine Morgan, now 61, also held the child's head under bath water, threatened to kill her and hit her in the face.
The abuse started when the child was just four years old and even though the victim is now an adult, she suffers nightmares and is 'haunted' by the memories.
The jury at Southampton Crown Court took five hours to convict Morgan of two charges of cruelty to a child under 16.
Morgan, who is a widow, will be sentenced at a later date, but the judge said she can expect to go to prison.
The victim, who cannot be named for legal reasons, told how Morgan told her moments before holding her head under water: "No one wants you - I'm going to murder you in the bath".
The victim added: "I have vivid dreams about when I was pushed in the bath. That's an incident I can always remember."
She recalled that she had been left with a 'burning sensation' in her mouth and 'felt a bit sick' after being force fed the bleach-soaked rag.
Prosecutor Elizabeth Bussey-Jones said: "She describes from a very early age being subjected to various forms of verbal abuse.
"She was told by Christine Morgan that she hated her, that she was a waste of space and she wanted to kill her. She recalls various acts of violence towards her.
"They vary from being pinned up by the throat against the wall, being smacked and having her hair pulled."
Morgan, from Romsey, Hampshire, denied all charges but told police in interview: "I would strangle her now."