If the speed of the electron is assumed to be constant, no net work can be done upon it, it does not lose energi and hence it continues orbiting. Since all the forces working upon the electron are perpendicular to its motion, no work can be done upon it, in spite of the continous change of the direction of the velocity vector.
A precondition for the result is the non-existence of any so-called magnetic field, i.e. only the Coulomb force has to be taken inte account. Elsewhere the author has succeeded in eliminating the very idea of "magnetic fields", basing the computation of electromagnetic forces solely upon Coulomb's original electrostatic law. (*)
(*) 'The Magnetic Force between Two Currents Explained Using Only Coulomb's Law', Chinese Journal of Physics, VOL. 35, NO.2, April 1997, Chinese Journal of Physics, pp. 139-149.