I would break the rest of it off and mill a thin piece of oak with tapers on both sides. Then I would stain it to match the floor and install it so it overlaps the tile a little. If the floors are a bit uneven, rabbit it so it fits flat.
You might find a ready made transition strip in the flooring aisle at a big box.
That piece likely has an inch wide leg under the tile.
You might find a ready made transition strip in the flooring aisle at a big box.
That piece likely has an inch wide leg under the tile.