We did not address that issue, which is why it shows up in the photograph. It's an old floor (many decades old) and we just left it for what it was.

We have the same broken tile floor on our patio. Some places need repair, but I'm not sure who to contact for these repairs. We are also in the Raleigh area. Do you have any suggestions for who we can contact?

Sorry, I don't have anyone to recommend--this kind of work is handled by the GC's. Perhaps try calling a tile supplier, like Best Tile, to ask about installers? It might take a couple inquiries, but I think that's your best bet (ha ha) for finding the right person.