The problem is those engines exert force on their enclosure in one direction, and the same force on the air in the opposite direction. That air, i.e. wind, will eventually slow down through contact with the ground, returning the momentum to the Earth.

This is going to be fun.

Let's say you are less ambitious that stopping the rotation of the entire Earth. You cut a small ring out of the Earth, a very tiny one: 100 km wide and 100 km deep. The circumference of the Earth is 40000 km, so the volume of this ring is 0.4 billion km3. The Earth's crust density is about 2.5 tons/m3, so we got ourselves a ring of one billion billion tons, or one billion Gigatons. This ring does exactly one rotation in 24 hours so that's 40000 km in 24x3600 seconds, that's 462 m/s, let's call that 500 m/s.

Let's unroll this ring. It has become a gigantic spear that travels through space with 0.5 km/s and weighs 1 billion gigatons. Our task is to stop it. We put a gun at the tip of the spear, and fire shells. The recoil of the gun will slow the spear down. The heavier and the faster the shells, the greater the recoil. The product of the mass of the shells and their speed needs to be equal to the product of 0.5 km/s and 1 billion gigatons. If the shells you fire have relativistic speed, you need to multiply with an additional factor to account for that, but we can't fire relativistic shells, so we'll ignore that.

Now you have a problem: obviously even the US Air force can't find one billion billion tons of stuff to fire into space. So we need to use less mass and higher speed. But the energy goes up as the square of the speed, as every kid knows (m v-squared over two). But hey, we have nukes, we can afford, right? Let's go crazy.

We can use ion thrusters. They shoot ions at truly mind-bending speeds. Up to 100 km/s. Two hundred times higher than the speed of our cosmic spear. So, we'll need to only fire 5 million gigatons of ions into space.

How much energy do we need? m v-squared over 2, everything in the metric system: m is 5e18 kg, v is 2e5 m/s, we get 1e29 joules. One ton of TNT has about 4e9 joules, so our energy requirement is about 25e18 tons of TNT, or 25 billion gigatons of TNT. The largest nuclear bomb ever in possession of the US Air force had a yield of 15 megatons. So, we are talking about roughly speaking 15 trillion such bombs.

But that is to stop only a tiny, tiny bit of Earth. A tiny ring that has a volume of 0.4 billion km3. The whole Earth has a volume of 1000 billion km3. You would need many quadrillions of Castle Bravo bombs to do the job.