I think Anne's possible health problems are mental more than anything : she is undervalued at home, also ignored, teated as the spinster aunt at mary's beck and call ad has never found anything-marraige,family, even intellectual pursuits- to divert her thoughts away from her lost love and what may have been for more than a few fleeting minutes at a time, it seems to me.

She has dwelt on what might have been for ages-years- following Wentworth's career at a very far remove...I think there is a good possibility she's more than a little depressed.