The best way to unkink and straighten an extension cord, is a trick on learned on the convention center floor during setup by electricians.
Simply put, you tie one end of the cord to a stationary object. Typically at a show this would be the cushman electrical cart, but any solid railing would do.
Then starting at the tie off point, you use a very large screwdriver, or a small length of straight metal pipe. Really anything easy to grip, but that has a long hard straight edge. A handle of a hammer for instance. You hold the screwdriver by the shaft, and put the plastic handle under your cord. The screwdriver/pipe will be perpendicular to the cord, as you pull the cord taunt, you fold it over the handle, and start to very firmly pull the handle along the length of the cord.
The process is similar to how one would curl ribbon using a scissor. It uses quite a bit of force to do, don't be afraid to throw your weight into it. I typically will form a "u-shape" in the extension cord, and place the screwdriver handle inside of it, and pull. Pull really hard while passing more and more cord past the screwdriver, stretching and flattening it as it runs along the handle. This should leave you with a nice straight cord. It is what is needed in order to tape a cord down along a floor.