Oh crikey - poor little thing.
My understanding is that your personality is set at about the age of 3. Up to that point, your brain is being 'wired' based on the experience you have combined with the genes you're given. But - just because I am compassionate doesn't mean to say that my daughter will pick up that gene, in the same way that I am not wired for maths (at all) but she is phenomenal at it!
Everyone learns from repeated behaviour they can witness. So although the mother may be a waste of space, there is the opportunity for this little girl to learn from others - certainly about love. You sound very special - she isn't your daughter and DESPITE her problems still want to be with her help her. Therefore you are CHOOSING her which, when she is older, she will recognise and will be a great basis of self worth.
My job is very very people orientated and part of it is enabling them to do well at interview. I work in a team of people and the psychologist we work with said that you can't change your personality, but you can (crucially here) modify your behaviour.
My daughter has some traits of aspergers or autism. Her aunt is gruesome in public (mouth open when eating, talks over the top of people, never turns up with gifts, reads bank statements and personal letters of other people etc) because she has never been told that any of the above are unacceptable, she has not been able to modify her approach. My daughter does exhibit some of the same issues, but I won't tolerate it - for her sake. So she has learnt to listen, to accept privacy - she doesn't quite get 'why', but follows the rules and is wildly successful in every single area of her life.
Your six year old sounds like she has an uphill battle. Genetically she may be pre-disposed to certain behaviours, but example is a fabulous way to ameliorate these tendancies or, if she is pre-programmed to be 'bad', then giving her coping strategies to enable her to suppress this behaviour will be the winning streak. Like learning a foreign language, it's not an overnighter, but it can be done. And I just cannot get over how well equipped she is to do this - with you in her camp.
No matter how hard - stay with it - it'll be worth it. My daughter loathed me until she was 11 - because I nagged her about the list above. I was prepared to lose her love so that she would end up able to function well as an adult, unlike her aunt who is a v v lonely and sad woman. At 11, my daughter matured perhaps, or maybe made the connection that I was actually FOR her not against. She is now the highlight of my day when she's home from school and she loves me totally.
xx