It sounds like the six year old may be "marking" trying to be dominant. She may feel "strange" due to the kidney problems and may feel that she's losing power over the other cat or whatever.

You might try spritzing a little Feliway on the mat after you've cleaned it and see if she'll leave it alone. Feliway is a very good deterrent for urine marking behavior. Just be careful not to get it into the box itself, or the other cat may avoid the box.
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Smile, it will make them wonder what you are up to!

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