A few pictures of how I stripped the wires for my columns on a grid-layout board, I used single core wire and a wire stripper.
Taking each piece cut to about the height of the grid:
Strip the end off in the normal way:
And for each following piece align them to about the width of a single key body:
If you trigger the strippers just until they almost bite you'll get nice clean sections of insulation cut down the core.
I threaded these behind the rows and aligned them into place before soldering, it's useful to slide the insulated sections as far away as possible from the solder point to prevent melting or neighbouring wires pressing through the through the warmed insulation:
You may be able to make out some small dimples in some of the pieces of insulation indicating their position while the wire was hot mid-solder, none of them melted all the way through but it's probably best to make sure they're repositioned so the row wires aren't sitting over them when in normal operation.