The Top-Of-Chimney Replacement Damper
So you have a damper that will not open or close, is rusted out or is non-existent. The economical solution is to have a top-closing damper installed.
The economical way to replace a fireplace damper
Typical fireplace damper-placed at the top of the fireplace (looking up.)
This fireplace damper is showing signe of rust due to lack of a rain cap on top of the chimney. To replace this exact damper would require cutting it out, then putting a new one in with mortar. This is a major act of surgery.
The top-closing damper mounts to the top of a tan-colored flue tile. A stainless steel cable runs down the chimney flue and connects to a bracket inside your fireplace. The cable is pulled down to close the damper. The cable is let up when you want the damper open. The purpose of a damper is to prevent heat loss when the fireplace is not being used. It also keeps out down drafts when closed. The top-of-chimney damper is on the right in the photo below:
This damper is made of stainless steel, so it will not rust. It also keeps rain and animals out. If a crown and flue tile is not present on your chimney (or if the crown and tile are in bad shape) then a new crown and flue tile will need to be installed on top of the chimney. See photo below to understand what a crown and flue tile are: