JT has a point. Living near the mountains, I sometimes take having an abundance of fresh water for granted. In fact, our local stainless mill uses 15 million gallons of processed fresh water a day for cooling steel while all residents of the county use 5 million gallons from the same source. It sometimes upsetting when the residents are ordered by law to conserve (like this summer) and the stainless plant continues to use its 15 million gallons a day without restrictions.
Try running the hose, just in the suspect area, not in all 6 or 12 gutters, for 5 or 10 minutes depending on the water pressure you have, in order to try to find the source of your leak.
Or wait for it to rain. However, you may have to stay up late if the next heavy rain starts at midnight or 3AM.