My water is icky. How long should I boil it for?

what you are seeing is sediment in your water. They flush the lines after repair work and the heavy flows disrupt sediment in the lines. Boiling doesn't clear sediment. It is safe to drink. In time it will clear up. If you want clear water for drinking sooner fix a pitcher put it out and let it settle naturally.

It's not sediment as it doesn't settle out.

Bringing water to a boil is adequate for disinfection. As soon as it boils its done.

http://www.high-altitude-medicine.com/water.html#heat