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Your mistake is that you did a second derivative of $L$, taking the derivative according to both degrees of freedom together. Instead you need to make a separate equation for each degree of freedom.

Since you have two degrees of freedom ($x,y$), you get *two* equations:

<p>\begin{align}
\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial\dot x} -
\frac{\partial L}{\partial x} &= 0\\
\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial\dot y} -
\frac{\partial L}{\partial y} &= 0
\end{align}</p>

The two equations you get this way are the equations of motion.

Another mistake is that you are writing total derivatives where you would need to write partial ones; the only total derivative is the time derivative. Since you are using them as if they were partial derivatives,  this is inconsequential in your calculation, though.