An asteroid is approaching at 70,000mph. NASA expects a decade of warning (other parts of the article say "decades or more"). In 10 years, that asteroid travels over 6 billion miles at that speed.
We're able to detect and track a 200 meter rock 6 billion miles away? That's way outside Pluto's orbit. And that's only a decade of warning.
We send out a mission at 70,000mph (Apollo only accelerated to 1/3 that speed), they'll meet only 5 years out.
Not quite right. The main concern are NEOs/NEAs (Near-Earth Objects/Near-Earth Asteroids) in orbits which potentially cross the Earth's orbit, which means that they are near to us and moving at orbital speeds similar to the speed of the Earth around the Sun. The detection process for "Earth-harmful" objects is a current project. Once you've found them, you can then project their orbital positions years into the future. It's feasible that one could be detected with a high probability of Earth impact years or decades in the future. It would be desirable to nudge such an object into an orbit with a negligible chance of Earth impact.
Missions have already proven that they can get quite close to asteroids or comets (at the end of its mission, NEAR was actually soft-landed on the surface of Eros). To move an asteroid, a propulsion device would have to be landed on the asteroid and turned on; slow thrust over time would adjust the asteroid's orbit. The main problem would be the gravity of a small (200 meter) object, meaning that it really doesn't have any. Eros was several kilometers long and wide, so it had enough gravity to allow NEAR to go into orbit around it. With a 200-meter rock, speeds would have to be adjusted exactly and then the approach would be extremely slow, to a feather-soft touchdown. That's pushing the technology. The just-launched ROSETTA mission to a comet has a lander that will show (in a decade) how easy, or not, it will be to do this.