what is her 2nd name ?

Marie Antoinette. AnnaSophia is named for her two grandmothers.

"I say,open this door at once! We're British !"

The "Marie Antoineyte" part was just a padding to fill anything in the details part, because it do not accept that part to remain empty ... I wrote that once and copy/paste it.

I wasn't sure whether annasophia is one compound name or 1st and second names ...

She treats it as one name, as she likes to be called 'AnnaSophia' not 'Anna', so I would take it as one name. Especially as it's spelt as one name with the capital 'S' in the middle. Even if it's derived from two separate names, it's up to each individual (and their parents) as to how their name is pronounced and spelt.
If there had been a space between 'Anna' and 'Sophia' then I'd go with first and second name (like Chloe Grace or Jenna-Louise etc), but there isn't.

She treats it as one so we should treat it as one.