I know that the magnetic field inside a circular waveguide is
How can I find the two components of the electric field, $E_\rho, E_\phi$ that is?
I have some doubts about your formula. It is my understanding that radial dependence of fields in circular waveguides is described by Bessel functions (see, e.g., http://www.eecs.ucf.edu/~tomwu/course/eel6482/notes/20%20Circular%20Waveguide%20and%20Coaxial%20Line.pdf )
EDIT (04/21/2013) I've just had a chance to look at Jackson, Classical Electrodynamics, 3rd ed. (please see @Thanos' comment below) The formula in your question is indeed approximate - it is used as a trial function in the variational principle. To find the components of electric field, you need to use Jackson's formula 8.26a, assuming $E_z=0$.