I have done enough "salvage" checkering, to be able to tell you, you should get your best 'Zen' feelings on, get set up in a comfortable and well lit area, and take your time at it, or you may regret ever having started!
As someone suggested in another thread on the subject, you could pay him a hundred bucks, and he'd kick you in the family jewels, it'll be cheaper in the long run, and less painful! LOL!
Trying to cut too much, too deep, and too fast, were the bugaboos. Best results came from carefully following the line, and as it became deeper and more cleaned up, move to the next, and repeat as require over the whole area to be done, paying particular attention to the ends of the cut lines, as over-runs are hard to hide. Figure on going over the whole pattern three or four times, rather than taking the line to depth before moving to the next one.
As per above, a multi line cutter, is only good if it exactly matches the existing lines. Otherwise, with each line being pushed over away or nearer than the original spacing, you will eventually reach a point where the cutter is trying to cut across the tops of the diamonds, and it'll leave a pretty ugly scar.
Kennedy's book shows a couple different styles of home made cutter. Another option, if you are at all handy as a metalworker, might be to head over to Youtube, look up a fella named Clickspring, and watch his video on making your own files from scratch.
A couple tools I found really handy, were three square and four square needle files. I bought the cheapest coarsest ones I could find. Three Square files are triangular, equal to a 60 degree angle. The Four Square files, are actually Square, and 90 degree angle. I broke the tapered tips off the ones I used for starting a line and for creeping up on the end of the pattern edge. With either three or four cutting surfaces at your disposal, they lasted a fair while before they got bunged up or dull, at which point, I just snapped off the last wee bit, and carried on.