If you do get her a motorbike/scooter get he a proper bike shaped one. One that you have to sling your leg over the seat, NOT a step through. The step throughs are not as stable as as a proper bike shape, there is nothing to grip with your legs (I always feel like a parrot on perch when on a step through) , a bit like riding a horse, its much better if you can hold on with something other than your hands.
If you got to a bike shop and purchase a new bike you will normally get the first year insurance and tax built in with the price and quite often they will throw in a helmet.
Your daughter needs her provisional licience and her CBT, but I think there may be more to do now as I passed my test a long time ago.
Persuade her that an open face helmet does not protect her pretty face if she comes off, an full face helmet at all times.
So the protective gear should be at minimum,
A full face helmet approx £60+ to £360
Gloves £15+ get the ones with armour on the knucles (proper bike gloves not something out of Primark) Nearly lost a finger because the glove shredded through.
Bike Jacket,£80+ they do really nice girls one, shaped for women. With Teflon fibres through the fabric, armour at shoulders, elbows and back also the extra pad at the kidney area. They are also waterproof.
Bike trousers,£60+ you can get jeans with the armour built in, or ones that match your jacket. Also leather ones with armour are good. But I have to admit that the jeans ones are more comfy in hot weather.
Boots, £60+ Always wear boots, no matter what. Ankles are very difficult to repair. I lost the skin from the top of my boots to my knee's in one accident but only had bruising on my ankles, the doctor said I was lucky, if I hadn't been wearing boots I may have lost my foot.
Also instil in her, SHE WILL COME OFF. Anybody who tells you they have never come off a bike is a lier.