Our steel kettle is all ick like that. Maybe I'll have to try that.
To clean copper that I work with I use a mixture of salt and lemon juice. I don't know about lime deposits though.
Be careful with vinegar and copper leaving it sit too long. Vinegar removes tarnish by eating away at the copper.
The vinegar's doing really well on the lime already. I was cleaning the outside of it too, with ketchup. That stuff is amazing. Are the insides of copper pots actually copper? Ours looks like it's coated with another metal on the inside.