Must we fast for two months as expiation for accidental killing, despite not believing that a light touch from the car caused her death, and not knowing her condition or place of residence? And if fasting is obligatory, how much should I fast, since I did not touch her with my car, but rather caused her to move backward?

If you were driving at the permitted speed and in the designated lane, then there is no blame upon you, because you did not directly cause the woman's injury, nor were you an aggressor. The rule is that liability falls upon the direct perpetrator, unless the indirect cause was an aggressor and the direct perpetrator was not an aggressor, in which case responsibility falls upon the direct perpetrator, not the indirect cause. Therefore, the liability for what befell the woman rests upon the other driver, unless he could not avoid injuring her due to his proximity to her, in which case there is no blame upon him, because the fault lies with the woman.

However, if you were an aggressor in your driving, such as exceeding the permitted speed, and the other driver was not an aggressor, then you are liable for what befell the woman, because you were an aggressing indirect cause. But you are not required to offer expiation unless it is proven that the woman died as a result of the accident, which is unlikely.