Does taking aspirin make my periods last longer?

No. No, it can make it slightly heavier but not longer.
It should not. Although Aspirin can thin the blood, it should not increase the duration of a period. In fact, anti-inflammatory medicines like aspirin, ibuprofen, or Aleve (naproxen) are commonly used to decrease bleeding, as they decrease the inflammatory respons in the uterus, and cause the blood vessels to close off some.