Calculating the capacitance
One way to look at this is as a R-C low pass filter.  The rolloff frequency is where the output has half the power of the input.  Since power is proportional to the square of the voltage, this means the output voltage is 1/sqrt(2) of the input at the rolloff frequency.

Unfortunately, your problem is at the half voltage point, which is the 1/4 power point.  For frequencies well above the rolloff, you can approximate the filter as attenuating 10 dB per decade above the rolloff.  However, you are not far from the rolloff point, so that approximation would have some error.  Simple RC filters are easy to analyze right at the rolloff, and then a decade in frequency or more either side of that, but not otherwise near the rolloff.  That means looking at this problem as a R-C low pass filter isn't going to help simplify things.

This means you have to do things out the long way using complex impedances.  What you have is a voltage divider, and normal voltage divider math will work except that the capacitor will have a complex component to its impedance.  In fact, the capacitor's impedance is all complex with no real component.

To solve this, do the voltage divider math to find the capacitance at which the magnitude of the output is half that of the input.

As a alternative to complex numbers, you could do this using phasers (magnitude and angle instead of real and imaginary).  The result will be the same either way.