Is the water which falls of the body (private parts) during istinjaa considered to be napaak ?

I have a urine problem where drops of urine tend to come out after urinating. Before I perform wudu I remove the tissue from the private parts. One day while at work I was about to pray salaat in the prayer room and when changing the tissue I noticed a speck of urine. I cleaned the urine with toilet paper and thereafter poured some water over the private parts. I did this in the sink. I only poured a tiny amount of water over the private parts and when pouring the water there was no urine on the body it was all cleaned with toilet paper. After cleaning myself I accidentally opened the tap with force and some water splashed on to the floor. Is the water that splashed napaak. It is possible that the water that splashed mixed with the water which fell of the private parts. I am worried and concerned that the prayer room might have become napaak. If water splashed on the body during istinjaa when using the western toilet how does one clean the splashes. If you wet your hand and rub the affected area with the wet hand will this suffice ?