bergs. Hope you've checked the motorised valve that diverts hot boiler water from the primary heating circuit to the indirect cylinder is working, not a motor failure, not jammed or clogged, and that hot water is actually passing into the tank. These valves usually have a manual over ride lever. Lots of info on YouTube, etc. Some valve motors can be changed without disturbing the pipework.
To clean the heat exchanger, I reckon it's acid you need to remove limescale. Try some lavvy limescale cleaner. If that works, brick cleaner (hydrochloric acid) should do the job on the large scale. There's probably a horror show in that tank. Dilute it down, it will eat limescale, the tank, and you! Be careful, don't want your drinking career to be shortened...... Next thing is pinhole leak from the bottom of the tank and bigger expense.
Swarfy.