Are there long-term effects of bladder infections on my child's health?

Possible. A single episode of urinary bladder infection will probably not have any lasting effect. However, recurring infections can affect a person's health, especially if the infection spreads to the kidney.
One or many? One not really, many yes. Stop caffiene, cut down on acidic beverages (juice and soda pop) as these both further bladder irritation, also make sure she's wiping from front to back and not reusing toilet paper. Potentially she could end up with a small bladder, a kidney infection, scarring if chronic problems.
Needs eval. If it is a single infection, then probably no long term issues. If they are recurrent, then he/she should have a full evaluation, including cystogram, ultrasound of kidneys and bladder, and possibly nuclear renogram. Need to rule out scarring of kidneys, bladder abnormality, and vesicoureteral reflux.