Don't people use straighteners for that?
I don't want her hair to melt. ;.;
Does your hair straightener/flat iron have options for low and high heat?
If so, setting it to a low and then using it, shouldn't cause the hair to melt.
Set it to low heat, keep the hair wet while you work, and don't keep the flat iron still in one place for long, and you should be quite alright.
It's scary at first but it can work absolute miracles.
If you're worried you could always try it on a pony you're more willing to test on.
Or if you're willing to test on this girl, test it on the end of a lock from the middle of her tail or something, that you can snip off and hide should it go wrong.