The chimney is deteriorating due to water getting in the cement block.
fix the concrete and use a construction adhesive like pl 400 or liquid nails
You are thinking only of the ones that attach to the flue, there are some that address your issue.
Do as the previous answer suggests by adding another flue tile. In order to be up to code you must use refractory cement to join the tiles. Also chip out a little)about half an inch from the flue and fill with a high heat pure silicone. As the flue tile heats up it will expand and then contract when it cools down. The silicone will keep the top from cracking and allowing moisture in. We have been chimney sweeps for 36 years and see crowns always spider cracking from the corner due to expansion. This prevents this from happening and allowing water in. Water will destroy a chimney over time It gets in through the cracks and makes its way to the smoke shelf picking up creosote and fly ash. The mixture turns to a lye like acid and eats the mortar holding the bricks together and soon causes the chimney to fail.
