I use Yusupov's system:
Pawn 1
Knight 3
Bishop 3
Rook 4.5
Queen 9
With many types of adjustments specific to the position.
I also value the Bishop pair at an additional .5. But really all of this gives you only a very rough idea of how things are going.
That the tactical value of a King would be 4 is a widely propagated misunderstanding. Probably caused by observational bias that when a King is useful, it is very useful. (But discarding all the cases where the King could not be involved, e.g. because passers were out of its reach, etc.) In practice the opening value of a non-royal piece that moves like a King is slightly under that of a Knight, as the OP suggests. (I.e. replace the Knights of one of two equally strong players by non-royal Kings, and he will score slightly less than 50% over a large number of games.) In the end-game it can be a little bit more valuable than a Knight, probably because it has mating potential (which would be important if there are very few Pawns).
GM Larry Kaufman has statistically analyzed how material imbalances occurring in GM games affected the outcome, and he found that a lone Bishop is almost exactly as strong as a Knight (both 3.25, although this seems a little bit dependent on the number of Pawns present), but that the Bishop that completes the pair is 0.5 Pawn more valuable.
Just looked at this Wikipedia page, and it appears many GM's value the pieces differently than their generic values (Bobby Fischer saying a bishop is 3.25, Kasparov dropping the rooks value from 5 to 4.5, etc.)
What do you guys assign to them? Here's mine:
Pawn: 1
Knight: 3.25
Bishop: 3 (3.5 when in a pair)
Rook: 5
Queen: 10
King: 3