It depends on what your goals are but generally yes, bi-amping will improve your sound.
In your case the Khorns were designed with bi-amping in mind, so you technically do not need an external crossover, but to realize the benefits of bi-amping you will need a way to adjust the gains of the amp(s) reproducing the mids and highs so if you do not have an amp that has adjustable gains, you probably will need an active external crossover. The frequency dividing function of the external crossover will be of little benefit for you since you will still be using the passive crossover in the Khorn.
One of the advantages of using an external active crossover is that you have the ability to set the crossover point for what works best for your room as opposed the predetermined crossover point that is hardwired at the factory.
One advantage of bi-amping a floor standing speaker is that you do not have to worry about any phase/timing issues.
Also keep in mind that you will be adding at least 4 more interconnect cables, and if you are running single ended, each of those new interconnects has the ability to pick up unwanted noise, not to mention that any noise picked up by the interconnects before the crossover will be amplified by the crossover gain stages, so if you have the ability to run balanced, you may want to consider doing that.