How to safely clean/remove recent mouse urine stain in carpet
Unless you are in an area that has some known level of serious infectious agent in the mouse population, such as Hantavirus, you should clean it immediately by hand with whatever detergent you have on hand. I would clean it 2x with hot water with detergent and then rinse the detergent out. Then dry it with a hair dryer.

EDIT

See Yosemite Hantavirus incidents

You could follow the rinsing with some sort of sterilizing agent like vinegar or alcohol, which would then have to be rinsed out. Note that these might leave an odor which would persist for some time.

EDIT

You could spray some (white) vinegar on the spot, let it stand for 20 min and then clean it. I think hand cleaning would be fine for one small spot. Honestly some parts of the country are infested with mice and disease transmission is rare.