Wet Kindles, like other wet electronics, can be ruined and never work again, or they can be salvaged depending on the damage. Water can "fry" electronics, making their circuits overload or short and never work again. Once this has occurred, intervention will not help. Amazon's one-year limited warranty doesn't cover water damage, so dealing with water damage immediately will help you save your Kindle.
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What Water Damage Does
Water is bad for electronics because it completes circuits that aren't supposed to be and damages -- sometimes even rusting -- small and delicate inner workings. When your Kindle gets a few drops of water on the screen, it's fine. When your Kindle gets completely soaked in water, the water may already have destroyed the device.
The first step after any Kindle water damage is to turn off the device. For just a few drops of water, wipe them off quickly. For a soaked Kindle, shake it a few times to dislodge water, then wrap it in a towel or other absorbent item and try to soak up as much water as possible. On the Kindle 1 and 2, slide off the back cover and remove the battery. Putting the Kindle 3 together again can be more difficult, so leave the battery in place.
There are a couple of methods for drying out electronics, so choose whichever is more convenient for you. Submerging the Kindle in white rice overnight will soak up excess water. A hair dryer on low (high could do more damage) will also dry out the Kindle. The important thing is letting the Kindle dry completely before turning it on.
Turning On the Kindle
Only turn on the Kindle when you are confident that it's completely dry inside. Even with precautions there may be damage. For example, part of the screen may not work.