How was he defeated if he's the strongest being?
He begged for mercy and got thrown into the Outer Dark where he currently dwells.
There is no contradiction: At the time he was finally cast in the void, Morgoth was not the strongest beeing anymore.
As explained in my other answer, it is important in The Silmarillion not to confuse Melkor, the mightiest of all except Eru, and Morgoth, which is the Evil Overlord form which remains once he scattered most of his original power in devious creatures and infused it in the very matter of Arda itself.
For Tolkien, the whole Earth became Morgoth's Ring, which explains why Evil always survives and prospers thanks to this marring in all things, and how he cannot be truely defeated until the whole world is broken and recreated anew.
Sauron with his One Ring reproduced the exact same process on a much smaller scale, since he was only a Maia to begin with: He only used it to gain influence over all the other rings, rather than over the whole Creation.