I dont know it to be a symptom of bulemia, but purple feet are a sign of poor circulation are her feet cold to the touch, are her hands and nose also cold or discolored?does she have tingling or numbness in feet or hands? If yes to any of these question all the stonger that she not getting good circulation Id get her to go to her general practioner.
Why is she homesick have you just moved house, is she eating properly , perhaps a visit to the Doctors to get her checked out first would be good, as she is thin .Are you aware of any thing that is upsetting her?