Yes the cone extractor is the tool to use

I am surprised that no other small loco enthusiasts have offered a view so I will try and add a small bit of help Nigel.

The two cones that you see in the picture will come out on some makes of injector and if they do so easily then fine. But if they are stiff then leave well alone. On some of the good injectors they are pressed in. You also have another, third, cone pressed into the body and you must not attempt to remove this as its depth position is critical.

Undo the top nut and take the ball out.

If the cones come out, do not try to clean them with any type of abrasive or metal object; delicate wiping with a tiny wooden or plastic blade to remove any film is the most you should attempt. Do not poke in wires. Inspect the tip of the steam cone, which is the most delicate, to see it is still a good order and not corroded or mis-shaped.

You need magnification. Look along the bore of the injector toward a bright surface and check that you can see a clean, round hole. Similarly check any removed cones, and again when they are fitted back in.

A used injector that has been fitted for a year or more will need a dilute citric acid (5%-10%) soak to remove limescale deposits. A couple of hours in warm/hot citric acid is good and an ultrasonic bath very helpful. Otherwise use a plastic syringe and tubing to flush it through during the soak. Be very careful not to damage a cone if they are detached or liable to fall out. I put a piece of silicone rubber tubing over the steam cone body to protect the tip.

If the injector has been contaminated by dirty water, especially well-fermented rainwater (!), the citric acid will not shift the muck and a strong alkali detergent cleaner will be needed, again hot and for 30 minutes or an hour. Do not use any strong acid that will attack the brass.