It depends: If there is no simulation relation, then there is also no bisimulation relation since the bisimulation relation would also be a simulation relation. Hence, if there is no simulation relation, you don't have to compute the greatest bisimulation relation since it follows immediately that there is no bisimulation relation.

If there is no bisimulation relation, there may still be a simulation relation, and even K1 may simulate K2 and K2 may simulate K1 in that case (see slide 42 for such an example).

If there is a bisimulation relation, it is also a simulation relation, but not necessarily the greatest one. So, you would still have to compute the greatest one that must then exist.

The particular intermediate steps cannot really be reused, except for the initial relation where state pairs with the same labels are constructed. That one is the same in these cases. Also the simulation diagrams are the same for all computations, but they are used in different ways.