I’m not sure the asteroids idea would save all that much energy, if any at all; to divert them into Earth’s orbit would be quite expensive, anyway, impacting enough of them would be extinction for most life on Earth.
We have to do it in a week or so, otherwise the longer nights are going to kill off a lot of life.
Just to make a small point: 4x10[sup]29[/sup] Joules is equivalent to 100 trillion megatons (1 MT is 4x10[sup]15[/sup] J) of energy, that is, the explosion of 100,000,000,000,000,000,000 tons of TNT.
This is a significant amount of energy. If your mechanism to re-rotate the Earth is anything less than, say, 99% efficient, then you are essentially exploding a trillion 1 megaton bombs on the Earth. That’s not conducive for life.
Go buy a house with a window that faces East.
Bah. Don’t listen to him - what you need is a giant propellor attached to one of the poles (or two, attached to both poles), and something to give these a precisely opposite spin. That’ll take care of the momentum without the need for an (external) fulcrum. Now, as to how you’re going to power it, I dunno. If you didn’t have the one week timescale, you could probably work out a way to use solar power, but as it is, that won’t work.
How about getting all the Chinese to start running east at the same time? This shouldn’t be much harder than getting them to all jump of chairs at the same time. (I apologize if you don’t know the old saw)
Well, assuming we have 1 billion chinese all weighng ~ 80kg. Thats 810^10 kg of chinese. E = 1/2mv^2t so we get v^2 = 10^19/t. Assuming a chinese person can run at 10m/s (about the fastest a human can run), we need 10^17s or 3 billion years of sprinting chinese to reverse the spin. Now, considering the fact that most humans can only manage 10m/s for about 4 seconds, I think some serious genetic engineering is needed.
Actually Some Guy is on to something. We can use the moment of inertia of the Polar-Wheel[sup]©[/sup] as leverage to apply torque to the world. Once we reverse the planet’s spin we launch the rapidly-spinning wheel into space and prepare to swap time zones. This of course is much more humane that launching Ring’s 1 billion Chinese into space.
Of course the whole process would severly disrupt the convection currents in the earth’s core leading to all sorts of earthquakes, and volcanos.
Obligatory link to “that old saw.”
Why expend the energy to reverse the east- west spin? Far less energy will be expended rotating the earth 180 degrees along the north south axis while leaving the present spin untouched. Then you’ll see the “sunrise in the morning”
So what’s a sunrise like in the evening? 
You mean flip the earth so that the south pole points towards the north star? That won’t work. The house will rotate along with the earth and you’ll still have sunrise in the same direction.
Besides, you need to transfer the angular momentum somehow. You need to transfer angular moment to or from something outside the earth. That means either hitting earth with high-speed projectiles at glancing angles (e.g. asteroids) or shooting high-speed projectiles (could be solid, gas, ions or any propellant) from the earth so they carry away angular momentum.
Rats
Do you think the Straight Dope Hamsters could be harnessed to perform this task?
Wouldn’t reversing earths direction cause time to go backwards? It seems I saw that happen in a movie once.
Sorry,. someone had to do it. 
If you manage to overcome all the hurdles and actually carry out this mad scheme, my guess is that surfers all over the world will be toasting you for many years to come.