I normally don't pull people from the same series to do a Random Who Would Win, but I just had to do this one! King Bumi is a badass and, arguably, the strongest Earthbender on the planet. Toph is innovative, forceful, and can bend metal, which is something that no other Earthbender on the planet can do. I can't imagine that these two would be enemies, so I would imagine the fight would come about as a product of Toph wanting to test her abilities against the Master of Earthbending, which I'm sure Bumi would be inclined to accept the challenge.
The fight would most likely take place in the giant arena where Bumi fought Aang, so there would be quite a bit of ammunition for both of the fighters. Bumi would, in my opinion, take an early lead, due to his sheer strength. Toph, though a powerful Earthbender, would most likely not be able to whether Bumi's onslaught if she continued to take it head-on. Toph would take to using her unconventional Earthbending style to redirect some of Bumi's attacks until she could figure out what to do. The benefit here is that Bumi would not be able to sneak up on Toph by going underground, because she can "see" everything in the ground through her feet.
This is how most of the fight would go and there wouldn't be much opportunity for Toph to use her Metalbending, because the arena appears to be entirely cut from stone. However, after a time, Toph would notice something as Bumi moved about... the vibration he put off having a familiar ring to it. As she got close to the old Master, Toph would reach out with her powerful hands and fingers and in the midst of the two's close-quarter combat, Toph would Metalbend the metal studs on Bumi's arm and leg bands to each other, maybe one arm to each leg or both legs together, before knocking Bumi on his ass.
Ordinarily, this would not be something that would stop the All-powerful King Bumi, but I'm fairly certain that once Bumi saw the young Toph's special skill, he would be so intrigued with it that he would immediately forget they were having a match and want to know more, thus making Toph the winner by default.
I honestly thought Bumi would win the fight, but the votes took it the other way! If Toph won the fight, it would most likely be in this fashion.
Hope you enjoyed the play-by-play!