Some thoughts...
Can she articulate why she needs to do this? Has someone made a remark that she's taken to heart, or is there some bullying going on - either to her or that she's aware of and wants to avoid being the target of? Are her friends the same?
When you tell her to hurry up, we're going out, what triggers the change from obsessing with the mirror to being ready to go? Running out of outfits, someone telling her she looks lovely, everyone else starting to leave, a certain time elapsing? Identifying this might help you find a strategy to reduce the time taken to reach the point of shifting to the next activity.
It would be easy to suggest she misses out on an event whilst she fusses, but judge this one carefully as she may blame her appearance rather than the fussing for missing out and actually become worse.
Have you tried letting her have timed access to a mirror once or twice a day? Let her fuss her appearance as much as she likes, whilst you ignore the behaviour, but once the timer is up, that's it, no more mirrors, until next time. Again, this might or might not help, you will know her and be best able to judge how she would react.