Actually, you have rediscovered the flaws of Rutherford's atomic model .
Now, in order to solve this, Bohr proposed postulates, one of them is:
Electrons while moving in a certain orbit cannot radiate energy and can only take pre-ascertained orbits.
What he said is based on the fact that Energy can't be radiated continuously. It is discrete in nature. If it had happened, we would have got continuous emission spectra as the electron would radiate at all the possible frequencies while collapsing with the nucleus. Thus, energy ought not to be continuous.
The electron can radiate energy to a certain frequency when it moves from higher orbit to lower orbit.
To think of it, suppose there is a ladder. There are certain points where there are steps. While you are descending, you have to place your feet at the next step, not in between. This is called Quantization of Energy!
Now, why is not energy continuous? This let the foundation of quantum mechanics. How did it start?
Let me brief.
It all started with blackbody radiation. Every body radiates EM waves due to the conversion of kinetic energy of the colliding molecules above $OK$ . Now, a perfect blackbody can radiate almost at every frequency. When classical physicists wanted to explain the blackbody radiation, they were in dilemma. They thought that the walls contain oscillators(they didn't know about electron) which would excite every frequency of the EM field. Now, since it is a container , the waves will interfere with each other to produce standing waves of several modes. Now, after reading wave mechanics, you will know that the high frequency waves will have infinite number of modes. So, in order to radiate at those frequencies, the oscillator has to give high energy to vibrate all those modes. Since, the modes are infinite, energy radiated at high frequency would be infinite!
This is the famous Ultraviolet Catastrophe !