As you know I think that Jack the Ripper communicated with the police. And I think he did so at least through 6 different sources, perhaps more.
Here are some examples of other serial killers communications with the police. And it could hypothetically give you a picture of what sort of communications I would expect from Jack the Ripper:
The Lipstick Killer of Chicago:
"For heaven’s sake, catch me before I kill more; I cannot control myself"
The Zodiac Killer of San Francisco:
"Dear Editor, This is the Zodiac speaking I am back with you."
The BTK killer:
"...How about some name for me, its time: 7 down and many more to go. I like the following. How about you? 'THE B.T.K. STRANGLER', 'WICHITA STRANGLER', 'POETIC STRANGLER'"
The Weepy-voice killer of Minnesota
called the police to say the newspaper accounts of some of the murders were inaccurate.
Happy Face Killer of Oregon:
A message was found scrawled on a wall at the Greyhound Bus Depot in Livingston, Montana: "I killed Tanya Bennett Jan. 21, 1990 in Portland, Oregon. I beat her to death, raped her and loved it. Yes, I’m sick, but I enjoy myself too. People took the blame and I’m free".
The Zodiac Killer of New York:
To The New York Post August 4th 1994: "Hi, I’m back".