Congratulations to her.
Worth saying first: the quotes you're seeing aren't a reflection of anything you've done wrong. New drivers are priced on risk statistics as a group, and that group carries the highest premiums of any age band, especially in the first year. It's frustrating, but it's the market, not bad luck.
A suggestions that may help:
Named driver doesn't work the way it's often suggested, and it's a bit tricky in her case. The usual trick is adding an experienced driver from the household to bring the average risk down, but that only works if there's someone with a clean licence who'll actually use the car sometimes. Since her mum doesn't drive and there's no other licensed driver in the house, that route isn't really open to her the normal way. Do either of you, as her grandparents, hold a full clean licence and would genuinely and occasionally drive the car, ask an insurer or broker whether adding you as a named driver would help. It has to be a real arrangement though, not just a name added to shift the price, and if you're not local to her or won't actually use the car it may not move the number much anyway.
Since black box is already in the quotes she's had, the next step is checking specialist young driver telematics insurers directly rather than relying only on comparison sites. Names like Marmalade and ingenie build their whole model around new drivers and sometimes beat the black box products offered by mainstream insurers. Direct Line and Aviva are also worth checking separately, as your instinct already suggested, since they don't always show up on aggregators.
Ask for an annual price alongside the monthly one. Paying monthly through an insurer usually includes a built in finance charge, so the true annual cost can be noticeably higher than it looks. If there's any way to pay annually, even using a 0% card rather than the insurer's own instalment plan, it's worth comparing the total cost both ways.
A broker is a good next move, as you've already suggested. They can access panels of insurers that don't appear on price comparison sites at all, which matters more for higher risk profiles like new drivers.
A few small things that sometimes move the price: parking off road on a driveway rather than on the street if that's an option, adding a Thatcham approved alarm or tracker, and double checking the annual mileage estimate is realistic rather than rounded up.
Check quotes in advance: Make sure that you put you need the insurance 15-20 days in advance. Not tomorrow or in a few days time. Insurers identify those that plan in advance as less risky than those that are last minute and you'll often see discounted prices for 15 days out.
Her occupation status may be worth revisiting. Once she's formally enrolled at college in September, it's worth re quoting with "student" as her declared occupation. Insurers weight this differently, and it can move the price in either direction, so it's worth checking rather than assuming.
It's also fair to expect this to ease somewhat after year one. A clean first year, especially on a telematics policy, tends to bring the renewal down. That's not guaranteed, but it is the general pattern insurers reward.