All elementary SI units, except the kilogram, have been redefined depending of a physical constant.
However kilogram still depends on the International Prototype Kilogram. Why is this? Is there no possibility of defining it depending on Planck's constant, for example, or some other constant?
This is essentially a duplicate of Why do we still not have an exact definition for a kilogram?, but the answers there are somewhat outdated. There is, in fact, a programme to overhaul the SI and make it wholly independent of physical artifacts; this programme is currently ongoing, having started in ~2013 and with a proposed date of 2018 for the formal redefinitions.
In particular, the kilogram will be redefined by fixing a value for Planck's constant (and subsequently also dependent on the definitions for the metre and the second, which in turn are given by fixed values of the speed of light and some (new) atomic transition).
For more details on this process, and what the new definitions will look like in practice, see What are the proposed realizations in the New SI for the kilogram, ampere, kelvin and mole?, as well as the Wikipedia page on the proposed redefinitions. For more details on why the kilogram was the last unit to get this sort of definition, see the proposed duplicate, Why do we still not have an exact definition for a kilogram?.
The short story for that, though, is well explained in Paul's link to this Nature News piece:
The kilogram is the only SI unit still based on a physical object. Although experiments that could define it in terms of fundamental constants were described in the 1970s, only in the past year have teams using two completely different methods achieved results that are both precise enough, and in sufficient agreement, to topple the physical definition.
Simply put, it turns out that old-school balances are surprisingly accurate, and that building experiments that are precise enough to beat them (and doing that in a scalable, metrologically sustainable way, with multiple independent routes that verify each other) takes several decades to do well.
