Extended Kalman Filter (EKF) for Non Linear (Coordinate Conversion - Polar to Cartesian) Measurements and Linear Predictions
Update
If I understood your model, you have a model of Constant Velocity in 2D (Cartesian Coordinate System).
While your measurement are in Polar Coordinate System.
Pay attention that your measurement function is:
$$ h \left( x, y, {v}_{x}, {v}_{y} \right) = \begin{bmatrix}
\sqrt{ {x}^{2} + {y}^{2} } \\ 
{\tan}^{-1} \left( \frac{y}{x} \right )
\end{bmatrix} $$
Hence you Jacobian becomes:
$$\begin{aligned}
{J}_{h} \left( x, y, {v}_{x}, {v}_{y} \right) &amp; = \begin{bmatrix}
\frac{\partial \sqrt{ {x}^{2} + {y}^{2} } }{\partial x} &amp; \frac{\partial \sqrt{ {x}^{2} + {y}^{2} } }{\partial y} &amp; \frac{\partial \sqrt{ {x}^{2} + {y}^{2} } }{\partial {v}_{x}}  &amp; \frac{\partial \sqrt{ {x}^{2} + {y}^{2} } }{\partial {v}_{y}} \\ 
\frac{\partial {\tan}^{-1} \left( \frac{y}{x} \right ) }{\partial x} &amp; \frac{\partial {\tan}^{-1} \left( \frac{y}{x} \right ) }{\partial y} &amp; \frac{\partial {\tan}^{-1} \left( \frac{y}{x} \right ) }{\partial {v}_{x}}  &amp; \frac{\partial {\tan}^{-1} \left( \frac{y}{x} \right ) }{\partial {v}_{y}}
\end{bmatrix} \\
&amp; = \begin{bmatrix}
\frac{x}{ \sqrt{ {x}^{2} + {y}^{2} } } &amp; \frac{y}{ \sqrt{ {x}^{2} + {y}^{2} } } &amp; 0 &amp; 0 \\
-\frac{y}{{x}^{2} + {y}^{2}} &amp; \frac{x}{{x}^{2} + {y}^{2}} &amp; 0 &amp; 0
\end{bmatrix}
\end{aligned}$$
The function of the measurement is the one connecting your state vector into the measurement and not the other way around.
Let's look on Wikipedia EKF Model:


In you case $ F $ is constant is the model is linear.
What's in Wikipedia called $ H $ is the $ J $ I derived above.
Dimension wise, all is perfectly defined.
Implementation
I implemented a general Kalman Filter Iteration with support for Extended Kalman Filter (With option for Numeric Calculation of the Jacobian).
I also added option to Unsecented Kalman Filter (UKF) Iteration, so you will be able to compare.
Here is a result in with the same model as yours:

The full code is available on my StackExchange Signal Processing Q51386 GitHub Repository.
24/08/2018:
I added UKF implementation which is pretty general so you could use it in various models.
Original Answer
Converting coordinate system is the main reason the Extended Kalman Filter was invented.
Let me give you a tip, it doesn't work well in those cases.
If you use Non Linear Transformation use something that will both make things easier and better (Yea, usually it doesn't work like that, but in this case it does) - Use the Unscented Kalman Filter (UKF) which is based on the Unscented Transform.
Once you utilize that there is no need to derive the Jacobian.
All needed is to apply the non linear function $ n $ times (On each Sigma Point).
It is easy to see that linearization doesn't work well for propagating the mean and the covariance in many (Most) cases.
The UKF directly approximate the calculation of the integration of the non linear function which calculates the mean and covariance.
It will make things easier as you'll be able to skip the linearization step and only know the coordinate transformation function.
In modern tracking we usually stay away from EKF and utilize methods which better approximate the integrals of the first 2 moments propagation.
The most common ones are the UKF and GHKF (Those are called Sigma Points Kalman filters).
Their generalization is the Particle Filter which in most cases is over kill.
Update
Have a look at EKF / UKF Maneuvering Target Tracking using Coordinated Turn Models with Polar/Cartesian Velocity.
From their conclusion:

We have shown a range of coordinated turn (CT) models
using either Cartesian or polar velocity and how to use them in
a Kalman filtering framework for maneuvering target tracking.
The results of the conducted simulation study are in favor
of polar velocity. This confirms the results of the previous
study [11] and extends it to the case of varying target speed.
For polar CT models, the performance in terms of position
RMSE of the predicted state appears to be comparable for
EKF and UKF. As the UKF does not require the derivation and
implementation of Jacobians it might be more straightforward
to implement. The RMSE provided by the Cartesian velocity
EKF and UKF turned out slightly worse. Interestingly, the
sensitivity of the RMSE with respect to the noise parameters
was decreased by using EKF2 and UKF in the Cartesian case.
This, in addition to the simpler implementation and lower
computational cost of UKF over EKF2 results in a recommendation
for UKF if Cartesian CT models are preferred.

Basically telling you, don't bother with Jacobians, just use the simpler UKF.
Another comparison is made at Implementation of the Unscented Kalman Filter and a simple Augmentation System for GNSS SDR receivers with:

As can be seen, UKF implementation does not require
linearization (state transition function and measurements
functions are directly applied to sigma points) and it can
also work in presence of discontinuities. The prediction
only consists of linear algebra operations. All such
advantages are fundamental for minimizing
computational load in an SDR implementation.


While the classical Kalman Filter implies the propagation
of n components for the state vector and n2
/2+n/2
components for the Covariance matrix, the UKF requires
the propagation of 2n+1 sigma points only.


Furthermore, UKF is more insensitive to initial conditions
with respect to EKF. It has been demonstrated that UKF
rapidly converge also in presence of an initial position
error of several Kilometers.