I have never refactored code to use strict; because all of my code has been written under strict (command-line one-liners excepted), but when I have had typos in variable names, I have always had great success by simply running the script at a terminal and letting perl point out the errors.
Since this is presumably functioning code, we can expect that there will be no variable-name typos. The simplest option for moving to strict would be to add use strict; and then use vars at top-level, listing the variables already used until all are declared. Narrowing the scopes to lexicals can then be done incrementally, using grep or Emacs' M-x occur to identify where each name is mentioned in the code and understanding of the program to determine where values can be reduced to narrower scopes.