Emkana -- I've done this, and have lots of Gm relatives, so no doubt more to come.
FWIW:
(1) blood relatives are one thing, friends of friends etc quite another. I've generally said yes to cousins (and I have a big extended family) but friends of relatives etc is a whole nother ballgame. Not okay unless they're friends of yours, or you happen to have a child the same age who is in need of an exchange themselves.
(2) unless you're geared up for teenagers, both you and the 15yo will be miserable. You're not in entertaining teenagers mode, plus you have no other teenagers handy that said 15yo can go out with to cinema/swimming/bowling etc. Even if you make shitloads of effort, it's still not going to be that much fun for anyone, and more to the point she's not going to learn much english anyway.
(3) 17yo+ is another ballgame cos you can put them to work in an aupair-type way, but I'm guessing that's not what they're offering?
(4) tell aunt all of the above, and tell her to get her friend to investigate some age-appropriate opportunities for the girl, viz summer schools for international students at UK boarding schools, EF-type language courses or UK activity holidays (PGL-type things) which also take non-english speaking kids for language-learning purposes.
Honestly, stay tough. We had a 19yo cousin staying for 6 months when ds was a few months old, and even that was a bit of an effort although she was ostensibly old enough to be useful around the house (ha!). Unless the 15yo is the oldest of a family of 8, in which case the equation might look a little different the whole idea will be doing you and her no favours at all.