Buy a bottle of store brand cheapo Sudsy Ammonia. For me it costs less than $2. Pour that in the kettle full strength and let it sit (cold) overnight. The next morning all that gunk will be loosened enough you can scrub it off. Don't heat the ammonia or mix with any other chemicals.

I'd strongly recommend using non-toxic cleaners on ANYTHING you put food (or water your family will drink) in. You could have gotten the same results by making a paste of baking soda and water, spreading it over the spots and letting it sit for a bit before cleaning it with a non-abrasive nylon scrubber. A cut lemon rubbed on the spots is also very effective (I've actually cleaned my oven with lemon--it's amazing. Please, next time, start with something that won't poison your family if by any chance any residue remains on the pot.

I was talking about the outside of the kettle - not the inside. I'd already tried vinegar, baking soda, ammonia, orange cleaner and noting would take off the gunk. It was like the gunk that gets on your oven walls. I have a glass top stove and wonder if that has anything to do with it. When I had a burner stove, the tea kettle did not get this brown buildup on the outside.

Thx for your great advice. It worked really well for me too. I use my large AGA cream enamel kettle on a gas stove, but I think the stains have got something to do with the size of the gas flame, as if it is set too high the flames lick up the outside of the kettle. I know that since we’ve reduced the flames beneath the kettle, we just don’t get these horrible stains any more. I’ve also stopped leaving it on the stove while cooking, just in case.

On the inside, just soak in heated vinegar, rinse well and repeat as necessary. It’s hard to see how scaled the inside of a kettle can get.