I would NEVER immerse the board in water, it might seep into the electrolytic caps and inside the switches. I think Ray's 'first' part was best. Holding under the hot tap, and scrub lightly with a toothbrush. I'm sure pee is easily washed off, it's water-soluble.
OTOH, the cat probably needs a bath anyways.
The reason that i recommended IPA is because I once ran a depot repair for Compaq PCs at one of the larger RBOC facilities. Compaq recommended IPA for spills on keyboards after flushing with hot tap water.
The scientific reason for IPA is that it mixes readily with water and reduces the water surface tension, therefore the water can be shaken/blown out more readily. I respect the fact that water soluble flux could be an issue since the old guide was based on rosin solder. Still, I think flushing with hot water, lightly brushing with an old toothbrush, and then using the IPA to reduce the surface tension before drying.
Here is a scientific link regarding IPA.
The water beads up on the penny and the alcohol spreads out flat. Many more drops of water can be added to the penny than drops of alcohol.
I guess in the final analysis, one must weigh all the issues and decide; one thing is known, pee is an acid and will cause long term problems.