Q: and what is the source of energy for the Earth's inner core?
Um, don't we know that already? I thought there was a shit load of radioactive decay going on down there?

And tidal heating. The interior does indeed have a tide, and when it moves it experiences friction.

Plus the friction caused by the convection currents.

Plus we're floating in a vacuum, so the planet is well insulated.

It's no surprise to me that the planet isn't cooling down in any hurry.