LEDs usually do not break, as these do not explode or break by falling on the ground. But if you intentionally exert a large amount of force, they may break.
And when they break, a care should be taken to dispose of them safely.
LEDs do not have mercury in them, but they have some other elements including lead, arsenic, Nickel, and Copper, that can be toxic in large quantities. Again, only in LARGE quantities.
Red LEDs has the most amount of Lead, while the white LEDs contain nickel.
But keep in mind that these quantities are far less than your car battery, a item that you use on daily basis.
So the point is that you should be aware of the fact that if LEDs breaks you should handle it with care and dispose of properly.
Use latex gloves and a mask, pick up all the debris with a broom, fill the waste in a sealed jar and then throw away any things that came in contact with the waste, like the bed sheet, even the broom.
Scientific American has an article on the Dark Side of LED. But if you read that article I would encourage you to read the comment section as well because that will give you the full picture.
SFGate also has an article that also mentions how to clean up broken LEDs.
Both of these articles mentions a research, as the bases for their articles. You can read that to get firsthand knowledge.
After all that, I have concluded, LED mostly do not break. And the toxic metals in there are in glass of LED, so treat it just like a glass waste. The information here is not to create a fear, but to give awareness about the latest research on this topic. LEDs are not a new killer 🙂