The energy of an electric current does not come from the speed of the electrons in a wire. It comes from the fact that they are pushed through the wire by a voltage difference, also called an electromotive force.
Think of a river turning a water wheel. Does the river have to be moving fast to turn the wheel? No. The river turns the wheel because the water is driven by gravity as it flows downhill. The pressure on the paddle from all the upstream water is what turns the wheel. The elevation of the source of the river is the source of energy.
In the same way, an electric current is driven through the wire by a battery or generator. Anything put in its path, like a light bulb, would be able to stop the electron flow if there weren't a voltage pushing the electrons through. That voltage is the source of energy--a potential energy, like gravity. The electric current is a means of transmitting it to the light bulb.
To contrast, the energy in a particle accelerator--where charged particles fly through a vacuum at near the speed of light--is kinetic energy. The work done by the beam at the end of the accelerator happens because of high-speed collisions. Each particle does work because it has high kinetic energy and is no longer accelerated by the time it hits its target. The particles in an electric current in a wire do work because they are pushed through the wire by the voltage. If the accelerator is turned off, the particles already in mid-flight will still do work upon impact. If a voltage in a wire is turned off, all the electrons will immediately stop and cease doing work.
The electrons in a wire never build up kinetic energy because the energy is immediately deposited in the device at the other end of the wires. In fact, the energy travels faster that then electrons. Even though the electrons travel at centimeters per second, an electrical signal travels at about 80% the speed of light, which is why a light turns on immediately when a switch is thrown.