Kidney pain is not something I have dealt with. It's unsettling that salted water would make your kidneys feel strange. Some ideas --
If you're trying to drink all your day's salt in one glass of water in the morning, that would be a very concentrated solution. Consider putting salt into all your water so the solution you drink is more dilute -- such as 1.25 g salt per liter water.
You could put the Lugol's into water with some dissolved Vitamin C. That would approximate taking Iodide only, to see if it's the Iodine portion that's bothering you. If that helps, you could take all your Iodine in Vitamin C and water, even mixed in with your salt water, if that worked for you.
The adrenals sit on top of the kidneys. They respond to Vitamin C and salt. Are you sure you don't in fact have adrenal issues, rather than kidney issues?