Dated honorific term for one's own or someone else's elder sister. In modern Japanese, お姉さん or お姉様 are more common.
姉君はどちらにいらっしゃいますか。
Where is your elder sister?
Compound of 姉 (ane, 'elder sister') + 君 (kimi, honorific suffix). The reading あねぎみ reflects a historical honorific form.