electromagnetism - The energy stored in the electromagnetic field of an electron

Due to Wikipedia the total energy per unit volume stored in an electromagnetic field is

$$u_{EM}=\frac{\varepsilon}{2}|\mathbb E|^2+\frac{1}{2\mu}|\mathbb B|^2$$

How does the energy stored in the electric field of the electron relates to its rest mass? How large part of the rest mass comes from this field?

And related, how does the energy stored in the magnetic field induced by a moving electron relates to its kinetic energy?

A Rigorous Derivation of Electromagnetic Self-force seems to give relevant information related to this question.