In order to complete this repair, you would need a complete damper throat and plate. We sell these products in varying sizes for different fireboxes, but the most commonly sold size can be seen here:
Vestal 36 Inch Cast Iron Throat-Style Fireplace Damper
In addition to the damper assembly, you would need castable refractory cement, such as this:
Rutland Castable Refractory Fireplace Cement - 12.5 Pound Tub
After the old damper has been cut out and removed, any remaining mortar will need to be chipped out and the space prepared for the new unit. A thin layer of mortar will need to be applied and the new assembly presses into position. Any extra space around the framework is then sealed up with additional cement.
Besides the cost of the damper assembly and cement, a typical contractor would probably charge $200-250 for this type of repair. Please let me know if I can assist you further.