Unfortunately the day fluctuates, and is actually lengthening over time: http://en.wikipedia.org/wiki/Fluctuations_in_the_length_of_d...

We need at least one unit of time which is fixed and unchanging in order to do scientific calculations, and we have chosen the second to be that unit.

So, instead of adding or subtracting leap seconds, we adjust the speed of light (and every other physical constant that includes a time unit)?

Also, which day would become the new standard: the solar day or the siderial day?

That would be a really terrible approach. Time is the ultimate basis for practically every other quantity we know how to measure. The SI second definition is literally the last thing you'd want to mess with.