The Coriolis Effect's contribution to wind speed is negligible
The acceleration due to the Coriolis force is $$\mathbf{a} = -2\mathbf{\Omega} \times \mathbf{v},$$ where $\mathbf{\Omega}$ is a vector with magnitude equal to the angular velocity of the coordinate system ($7.29\times10^{-5} \text { rad/s}$), and diretion equal to the axis of rotation--that is from the Earth's south pole to north pole. $\mathbf{v}$ is the velocity of the air in question.
The Coriolis effect is a fictitious force, that is, it is not a real force acting on an object but it is an artifact of calculating motion in a non-inertial frame of reference. In order to estimate the effects of the coriolis force on a hurricane, we can relate this Coriolis force to another fictitious force, the centrifugal. The centrifugal force is the important (fictitious) force in keeping objects in orbit by balancing out the pull of gravity. We can set up an analogy for how a hurricane works by setting the Coriolis force equal to the centrifugal force for a theoretical air particle rotating around the edge of the hurricane, to see how fast an air particle will rotate if its motion is dominated by the Coriolis effect.
The centrifugal force is $$\mathbf{\omega} \times (\mathbf{\omega} \times \mathbf{r}),$$ where $\omega$ is the angular velocity of the hurricane (note the omegas in the equation should be bold vectors, that is not showing up in formatting), and $r$ is the distance from the center of the hurricane. We can drop the vectorization by assuming particle motion is in a circle on a plane with axis of rotation of the hurricane perpendicular to that plane.
The velocity of a rotating object is $$\mathbf{v} = (\mathbf{\omega} \times \mathbf{r})$$ which is looking familiar from our centrifugal force equation. For the Coriolis force, I argue that the result of the cross-product, in order to be perpendicular to both $\mathbf{\Omega}$ and $\mathbf{v}$, is in the same direction as the centrifugal force vector. Therefore we cancel the $(\mathbf{\omega} \times \mathbf{r})$ term, leaving only the $\sin{\theta}$ for the angle between the Earth's axis of rotation, and a plane on the planet's surface. $\theta$ is the latitude; the factor is 0 where the angle is 0 at the equator and thus there is no Coriolis effect, and 1 at the poles. We solve for the magnitude of the $$|\mathbf{w}| = |2\mathbf{\Omega}\sin{\theta}| = 0.73\times10^{-4} \text{ rad/s}$$ for a latitude of 30 N.
What does this number mean? If we multiply that rotational speed by a distance from the core, we should find the component of expected wind speeds caused by the Coriolis force. Thus if we multiply by 100 km, we get expected winds speeds of 0.73 m/s.
Conclusions
First off, we can point out that if we use 1000km, we get expected wind speeds of 7.3 m/s, which is pretty significant. However, this is only a rude approximation, hurricanes have to follow the rules of fluid dynamics and such as well. The eye of a hurricane extends for many kilometers to either side, so this approximation is not useful at all at smaller distances, and hurricanes don't have enough energy to drive winds that fast out to 1000km (at least not on Earth). In real life, the maximum wind speed is sustained at the eyewall, at the boundary of the eye. The radius of maximum wind is around 50 kilometers.
At distances of 50km and up, our above equation gives Coriolis-caused wind speeds of less than 1 m/s. Obviously, a hurricane is much more powerful than this. A category 1 hurricane on the Saffir-Simpson scale has sustained wind speeds of 33-42 m/s.
Overall, the ability of the Coriolis effect to hold together a hurricane is small, around 2 orders of magnitude lower than the main contributer, which is the pressure gradient caused by rising hot air. From this Wikipedia link you will see that the Coriolis effect is a pre-requisite for cyclone formation. There needs to be sufficient force to develop the smooth, spinning, fluid flow in the atmosphere; a turbulent flow to the low pressure point will prevent cyclone formation.So, while the Coriolis effect is critical to the formation of a hurricane, it is not nearly powerful enough to cause 35 m/s + winds to spiral into a compact cyclone.
Given this, we can assume that we would need roughly two orders of magnitude increase in Earth's rotational velocity before the Coriolis effect can significantly affect the size and wind speed of a hurricane.
