1. First of all the contents of a WILL takes effect after the death of the testator and should be attested by two witnesses.
2. The question of challenging the WILL arises only after the death of the testator and the testator can change the contents of the WILL, write a fresh WILL any number of times and the last WILL will only be considered after the death.
3. Whether your mother has already executed the WILL in favour of your brother only?.
4. Why can't you convince your mother based on the points you have mentioned in the letter like your help during her health related issues and also your looking after her during crisis and convince your mother to share her property to you also, alongwith your brother, when she executes the WILL.
5. Even though your mother is not mentally ill, according to you, she suffers from a personality disorder and is emotionally unstable, do you have any evidence of medication relating to her personality disorder and unstability, like medical prescription, medicines used, hospitalised for such reason, etc. so that the same can be used as evidence in future, in case of you want to challenge your mother's WILL after her lifetime.
6. To conclude, I sincerely feel that somehow you should convince your mother to share her property to you also, by way of Gift Deed during her lifetime or through execution of a WILL, which takes effect after her lifetime.
7. In case your mother does not execute the WILL during her lifetime and dies intestate, then all the legal heirs, i.e., your father, you & your brother will get 1/3rd share in your mother's self-acquired property.