how do you properly pronounce the word "Annesley"?
It's not a word in the sense that you could find it in a dictionary; it's a proper noun (that is, a name).
It might be:
a family name (from Ireland, apparently), or
a village in England.
I haven't been able to find the pronunciation. If it's the family name, then the correct pronunciation is whatever the family members say is the correct pronunciation; and if it's the village, the correct pronunciation is whatever the inhabitants say is the correct pronunciation.
Sorry I can't help you more than that.