The part of the blade closest to the hilt isn’t sharpened as much as the tip is and some longswords dont even have an edge for the first foot of length which makes it easier to grip and better for blocking blows. This does not really hinder performance as straight blades do most of their cutting with the last foot of blade as opposed to curved blades which use more of the length in draw cuts.
To address the OP:
gmarkstephens brought up a good point in questioning the arming of the combatants, specifically the horse. He was, however, mistaken in stating that samurai were primarily foot-soldiers. While they trained for ground combat a samurai was expected to start the battle on a horse, and stay that way if possible. Samurai, like knights, were wealthy people in the community and quite capable of such a luxury.
Consequently, it could be expected that were a knight and a samurai to encounter one-another it would be on horseback. Broadsword vs. Katana is not a terribly sensible pairing in these circumstances; the knight would almost certainly have his lance ready for such an occasion. The outcome of Lance vs. Katana is clear. However, a respectable samurai (as they all were) would also have a weapon specifically designed for horseback, the mounted bow. This is a weapon the English never did perfect, but one the samurai had mastered. I am unsure of the ability of such a bow to penetrate armor, but after felling the opposing knight’s horse the samurai could afford to wait for an opening or just trample the poor man. This is all assuming the samurai didn’t opt to plink away until he hit a seam.