If her mom knows anything about tennis in the first place though, she shouldn't be doubting your ability.
Look, with all due respect, I don't care how good a 13 year old is, he/she will not beat someone who is 5+ years older than themselves (who has more experience).
A 14 year old kid in high school tennis (not in tournaments obv) was the no. 2 player for a quickly rising team in our state. A blue chip of the same age is their no. 1.
The matchups in the playoffs came down to this:
no. 1 singles: 5 star senior vs blue chip 8th grader
no. 2 singles: 4 star junior (me) vs 5 star 8th grader.
They came in thinking they would win. In the end, none of them won more than 3 games.
Just an example. Especially a girl (no offense) who is 7 years younger cannot come close to beating you.
So final tip: If she's still alive in the sectionals, make an effort to warm up with her in the morning, and give her strategy tips (a plus if her parents are right there). Make sure to be encouraging too... maybe she would like that enough.
I would also try to start out gradually. She still keeps her current coach, but maybe you could do like weekend clinics or something like that instead of trying to be a full time coach right off the bat.
Babolat Aeropro Drive w/ Solinco Tour Bite, 4 star recruit