The nits will never wash away. If she's still finding "hundreds of nits" it's either a reinfestation, or she hasn't removed all the old ones. And she can't count on old, treated nits to not be viable - I made that mistake early on. You HAVE to comb them out and remove them to be sure.
Nits will NEVER wash out of their own accord - they are glued onto the hair shaft with something very unique. Scientists and manufacturing companies have been trying to crack that formula for years, because if they can find either a manufactured version of the glue, or a solvent for the glue, either will sell a million. Even Supaglue will dissolve in acetone, but this glue won't. And that's from husband - a professional expert in chemicals and glues. if the nits are not manually removed, they will stay there until they grow out far enough to be removed on her next trim - several years. Of course, they won't be viable past a few weeks, or SUCCESSFUL killing with pesticide - whichever comes first.
Sounds to me like her mother is putting the chemical treatment through and not doing the full comb-through properly, just combing once through to see if she's got any, then rinsing out. A sure way to keep an infestation going... because of course, the bottle won't tell her to change direction and comb through again. Why should they? The more reinfestations, the more treatment they sell.
How often does she visit you? Because if it's weekly, you could manage to remove them during her visits to you. A repeat treatment in 7 days would show some half-grown lice, but shouldn't show full adults & loads of eggs, unless she's picked up another load. Which sounds like is what's happening.
So why does the mother blame you? And why isn't the school excluding her? In Australia the kid would be banned until all eggs, dead or otherwise, have been removed.
She's 5. It's an age where they make a lot of close physical contact with their friends. They have dress-ups in the classroom (and I bet her mother hasn't notified the school, or asked them to keep her away from dress-ups - that would alert the school and maybe insist on her being excluded).
Does she have an aide? If so, then there's a chance of the school maybe noticing sooner and doing something, unless her aide is short-haired.
At 5, little girls hug their best friends. Mothers let their little girls wear their hair out, or in cute little pigtails. Kids get their heads really close together.
Unless this is dealt with properly, this is going to be an ongoing problem until wither the main vector is found and treated, or until she gets older and there's less contact (about 8).
All you can do is be vigilant, and patient. Check her on arrival and if the mother has recently treated her hair, then do it again but without chemicals, just use conditioner and see if you can trap or remove any lice or eggs. At the risk of grossing you out, you can tell if an egg is live or not by seeing if it will crack between your nails. And sorry, you need to know. If she arrives with live eggs, then she is bringing the problem into your home, not catching it there.
What a tangle! I feel for you, trying to handle this one with diplomacy.
Marg