Perms don't last well, either; they fall out on the top faster than in the back, leaving kind of a funny-looking hair style. I'd see if you can't interest her in something else, like a new haircut that works with her straight hair and looks fresh and stylish.
Personally, I would never perm my 5 year old child's hair. I don't think she is old enough by any means, and the harsh chemicals will damage her hair for sure. I would focus more on teaching my child to be happy with what she was blessed with, whether it's her eye color or hair color, hair type, body type, skin color, etc. We are all so very different, and we need to learn to love ourselves. If you enforce this thinking at a young age, you may save your daughter from a lifetime of heartache and self-loathing.