Mermaid dd didn't recieve her NI number either! Like your dd, she didn't actually realise until she'd applied for, and got a job but then couldn't actually work there until her number arrived! They told us it would be up to 3 weeks but still had t arrived after 4. When she called again they said they'd have to start the process again so would be another 4. At that point I took the phone and 'insisted' that they speed it up. Her letter arrived within 3 days! So I would recommend a strongly worded phone call if you haven't already!
Dd has to be in for morning registration but after that can leave when she likes as long as she signs out.
So far she has chosen to stay in school (mostly I think because she enjoys the privilege of using the brand new super swanky 6th form centre/study area - and the snacks available in there . . !)
Fantasy dd's head of 6th form also insists on calling them 'non-contact time'. Dd says she's never heard anyone else, including teachers, refer to them as anything other than "frees"! 
Assessments happening in most subjects this week for dd as well.
She found out today she's been entered into the Maths Olympiad for Girls - think it's next Tuesday. Anybody else's dd doing it? Is it like the maths challenge?