The "shiney insides" might be an enamel coating - especially if it extends all the way up inside the pot and not just to the level of the sludge. The source of the shiney is going to make a difference in how you clean it.
Are there any markings to indicate where it was made?
For the spout - you're going to need a bottle or test tube brush.
"It ain't what you don't know that gets you in trouble. It's what you know for sure that just ain't so." - Mark Twain
Yes Michael, I bet it is enamel now that you mention it. It has the same feel as the inside of my LC French oven.
OK I just went to the website where the pot came from (hitting myself in the head for not doing that first) and found my tea pot. Michael, you are correct that it is enameled. It does not say where it was made, but it is a Japanese pot, at least in style.
What about baking soda and vinegar. The bubble action has cleaned many a sippy cup of milk left under the car seat from my nieces. It's a chemical type reaction, but it's safe. We also use this as a natural way to clean steel pipes, plastic pipes, copper pipes, a lot of things where you don't want to create any erosion. I've never had baking soda scratch.
Generally you can use salt and ice chips to clean seasoned cast iron, but I don't know about that enamel inside. I don't know how enamel handles salt. We use the salt and ice method on coffee pots where you can't use soap and it doesn't scratch the glass.