If your budget for a Kynar stripping tool is $0, here are two methods. I have used both methods and they really work! These are not my videos.
Here's a close-up of my modified nail-clipper. The donor needle is shown at left - note how it too gets deformed (note nicks near tip). A couple suggestions. Start from the tip and work up the size of the hole. Obviously you are trying to make the hole a tad more than the 0.01" wire diameter, but less than the ~0.02" insulation diameter. Unless you have some insanely stiff steel needle you won't be able to make a perfectly round hole. When you finish making the hole, I suggest you "de-burr" the edges so as not to scratch/scrape the bare wire as you pull the insulation off.