RMS current of a circuit driven by a white noise voltage source
It doesn't really make sense to talk about a single or "effective" value of \$X_C\$ in this context, since as you say, you don't have a single frequency that you're dealing with.

Instead, you need to take a step back and apply a more fundamental relationship:

$$I_C(t) = C\frac{d V_C(t)}{dt}$$

Since you're the mathematician here, I'm not going to tell you how to differentiate a random signal, other than to make a few qualitative comments.

A differentiator functions as a high-pass filter that deemphasizes low frequencies and emphasizes high frequencies. Therefore, if your voltage source is "white" (equal amplitudes at all frequencies), the current value will be very "blue", with arbitrarily high values at arbitrarily high frequencies.

The RMS value of this current (relative to the RMS value of the voltage) will depend on exactly what bandwidth you're considering.