USB to PS2 converter

I recently posted about my dead Contour optical mouse, whose cable had a
broken wire--one of five. Given the way the five wires were crimped into a
piece of plastic, it looked like it would be impossible to fix.

I also, however, have several old PS2 Contour mice whose rollers had gotten
clogged with cat fur. It turns out that some of the cables have the same
five-wire configuration.

I'd like to try connecting the unbroken PS2 cable to the optical mouse. What
sort of USB to PS2 connector will make Windows XP recognize the mouse as
Plug-and-Play? Or is that even possible?

Thank you!

Jo-Anne