Usually repairing the brick and mortar is not possible. Older chimneys may or may not have a clay tile liner. Any damage or breach in the flue liner may have serious consequences (fire or carbon monoxide escaping into the home). The hard part is simply reaching the damaged areas in question. I suggest:
1. Complete flue inspection by qualified chimney sweep will probably include a camera for a tall chimney.
2. Get their recommendations for repairing any breach vs. relining. Sweeps tell me with a flue of that age, 80 to 90% need to be rebuilt or relined. Most sweeps I know of want the chimney to meet today's standard for safety and proper function.
3. Look at cost vs. benefit, rebuild vs reline.
4. Installing a liner designed for the purpose is quite common. Expect to pay a couple of thousand bucks each. Cost varies and depends on height, condition of firebox, damper, accessibility, caps, etc.
You want the new chimney to last the next 100 years.
One other option is to install ventless gas logs. They don't require the use of a chimney. Properly installed, you get instant on and off convenience, heat efficiency, nice look. Down side is the flame can look a little artificial (a lot of blue, not so much yellow) since the ventless flame has to be precisely adjusted so it doesn't produce CO.
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