I too kicked up a hissy fit at the thought of going on holiday to Disney when I was around that age. I was adamant I wasn't going, but mums being mums I was given very little choice. I just couldn't understand why I was being dragged to a place that was designed for little kids.
BUT.. and here is the big but. I absolutely adored it. Admittedly I thought Epcot was rubbish (I'm a convert now at 23 though) But I did love Magic Kingdom and MGM.. I also wasn't so big on the shows, but I got dragged round them anyway.
The real swinger for me, however, was Universal studios. (there was no Islands of Adventure then) I adored this park to the point of begging to go back solo while my parents went off shopping. The parks are so safe my parents didn't mind at all.
So if you have the time maybe factor in a Universal trip, imo it's a haven for teens. Also show her pictures of the parks and she may be somewhat swayed seeing the Hulk Roller Coaster and stuff. It just looks cooler than Disney (in a teens eyes) less princesses and pixie dust, more marvel and RAWR!
I bet once you all get there she'll be the one in tears because she doesn't want to return to rainy old England.
Failing that, I'll come with you. Hahaha.