Don't know if the OP will ever return, just lurk here or whatever but...
I wholeheartedly vote for NO RELAXER
I think your daughters' hair looks completely MANAGABLE
the way it is now. Really, it does.
With the proper products, techniques, and MINDSET
, perhaps you and your daughter can one day view her hair that way too.
First, stop hating her hair. Please. Just do that, okay? No judgement against you. BUT... No matter how much you say you love your daughter, and I believe you do, and no matter how much we all say hair is just hair, hair is tied into self esteem because it's one of the most accesible things to see of ourselves on the daily. So, you and your daughter have to learn to love what she has, big hair and all.
Second, realize that ANY hair type can take some degree of time to take maintain. I have a daughter with basically str8 hair (she has some waves) that is Asian-like in silkiness and it still takes much time to do her hair because it tangles on itself at the ends. But, it's easier to do her brother's hair and he has corkscrew curls that matt and frizz. Go figure.
Third, check out the resources some of the other posters linked to, especially BeadsBraidsBeyond.blogspot.com (a huge wealth of hair care info and hair style ideas there!)
and Tightlycurly.com. I used Tightlycurly.com site when I first noticed that my daughters' hair was not growing longer. While I never put any chemicals of any sort in their hair, I didn't understand proper hair moisturizing and their hair was basically way too dry and didn't go much past shoulder length for years. The #1 thing that got my daughters to all have hair that now comes past their bottoms was learning to moisturize properly.
All I have to do now after washing their hair is put Virgin Coconut Oil on their damp or wet hair and braid it or put it in a ponytail and their hair stays moisturized for several days. That's it. Took 2 years to get to that level but it was worth it 100%.
Fourth, relaxed hair, while it certainly will be easier to comb when dry and easier to kept less tangled when worn down, relaxed hair has its own set of proper maintanence techniques that you'll have to either learn yourself or pay someone to do. It's not all a walk in the park. The chemicals used it relaxers could eat away your daughter's hair and burn her scalp. Relaxers can and have caused permenant hair loss, not only for people who didn't do a relaxer right, but for people who for years had thick relaxed hair that was done correctly. Some people get hair that seems to thrive with relaxers and others get alopecia hair loss. Huge gamble, like with most things.
Take the time to learn all you can about relaxers, and don't let the Kiddie Relaxer label fol you. Kiddie Relaxers and Texturizers have the same chemicals as regular relaxers but in different quantities. Still, it all equals the same stuff that is powerful enough to dissolve soda cans and burn skin, not to mention that constant inhalation of the chemicals can cause adverse reactions in certain people.
Time, patience, and proper understanding of what your daughter's hair really needs is all it takes to get your daughter on the right path to being able to enjoy her hair.
Now, if you do decide to relax her hair...for the love of all that is right in the world please, please, please find someone who can do proper relaxers & relaxer maintanence, and who has extensive experience with relaxing and taking care of children's relaxed hair
. If you can, take a look at other her/his other child clients' hair. If any of them have missing edges, bald spots, wafer thin hair, traction alopecia, or super dry hair...RUN, RUN, RUN! Also, check to see if MANY of that person's child clients have hair that comes past their shoulders. If all the stylisty can do is maintain short hair, RUN, RUN, RUN. That stylist is likely extremely scissor happy and will give your daughter hair cuts instead of trims, or that stylist is one who can't contribute to hair health well enough to help your daughter retain hair length.