In short, based on the forensic facts, an Irish-American "Prince of Quacks" called Joseph Tumblety, a mysogynist, was the police's only suspect until they lost him when he skipped bail to France and the USA. When he died in 1903 he was carrying two rings identical to the ones stolen from the second victim, Annie Chapman.
It seems the police decided to forget about him and look elsewhere so as to avoid embarassment.
The case was the beginning of what's now called tabloid journalism. One of the most famous elements in the story is the letter from "Jack", which was actually forged by two journalists.