If an electron were really spinning round in circles, it'd rapidly shed all its energy as electromagnetic waves (radiowaves/light/X-rays etc).

If you want to do a purely hypothetical calculation, you could consider the radius of the electron orbit (let's start with hydrogen - look it up; call it the atomic radii), work out the attractive electrostatic force for +1e and -1e (look up the formula for electrostatic attraction), equate that to the classical centripetal force F=mv2/r or mw2.r and there you are (look up m, the electron mass). Basic first-year 'A'-level stuff.