@Batzi
Who says that the core has to always stay in his head. Maybe he can move it around like Melzargard.
As for the ki, he has the androids absorbed and they have unlimited sources of energy.
Now I understand how Cell survived and became even stronger. @Batzi and @Dj0rel explanations make perfect sense! Cell has been ki blasted through his body many times so it is possible for his core to have survived the explosion and since he was closer to death than any other saiyan ever has, it makes sense for him to have experienced growth way beyond Goku and Vegeta after their growth after death. For example, on Namek, Goku nearly died as no threat against Freiza first form but got healed and was able to keep up with Freiza at fourth form which is a HUGE power up. And for the amount of ki regenerating would require, by using the self maintaining ore of Android 17 within him, Cell has an infinite supply of ki long as he is supplied a little time to get it all at once, and he did get all the time he needed to completely regenerate.
Also, @gormadoc and @Cavery210 King Kai can lie and be cocky, you know that right? And in the west part of their universe, universe 7, King Kai is the most important person so their is an element of truth.