In an adult six months of growth averages three inches, could well be different in a young child. The demarcation is your guide to when the relaxer has grown out.
Not sure what you mean by have you permanently ruined her hair? If you mean will the straight parts revert to curly the answer is no. If you mean will it eventually all grow back curly the answer is probably.
Safer is a relative term so not the same as safe: relaxers/ texlaxers and curly perm solutions are highly alkaline so damaging to skin and hair which are both naturally acidic.
Maybe teach your little girl to love and appreciate her natural youthful beauty? There are plenty of curly groups and pages on Facebook and other social networking sites that celebrate natural hair.