What is the actual exhaust velocity of the exhaust from the J58 engines at cruise (Mach 3.2)?
It's not a cut and dried single figure but  this 
SR-71 PROPULSION SYSTEM - P&amp;W J58 ENGINE (JT11D-20) seems to have enough information to allow you to arrive at a reasonable figure with 'some work'. If I had to choose one information source to get a feel for the answer this would be on the short short list (so far :-) ). Unlike most other sources, the result will be at least somewhat meaningful, if not overly accurate.   

It also seems to make it 'clear enough' that if someone gives you a simple unqualified numerical answer it will be meaningless, or wrong, or both :-). It's a 44 page PDF slideshow - but has enough detail to show how different the J58 is to what you may expect. 

He also supplies these references which will almost certainly be interesting, and possibly also useful. 


Connors, Jack, The Engines of Pratt &amp; Whitney: A Technical History, AIAA (American Institute of Aeronautics and Astronautics, Inc.), Reston, VA, 2010, pp. 321-332.
St. Peter, J., The History of Aircraft Gas Turbine Engine Development in the United States: A Tradition of Excellence, ASME, New York, 1999, pp.264-275.
Campbell, David H., F-12 Series Aircraft Propulsion System Performance and Development, AIAA No. 73821, 1973. Also found in AIAA Journal of Aircraft, Vol. 11, No. 11, November
1974, pp. 670-676. Note: This paper presents information only up to Mach 3.0 due to security restrictions at that time.
California Institute of Technology, Course Ae107, Case Studies in Engineering: The Lockheed SR-71 Blackbird, Spring 1991. Published by Cal Tech for course students and presenters only.
Anderson, J. Thomas, unpublished training course syllabus, How Supersonic Inlets Work: Details of The Geometry And Operation Of The SR-71 Mixed Compression Inlet, 2013.

If you do want a single misleading and probably wrong figure :-) then this book say 18,587 m/s at Mach 3.2 (table 1 page 175)


More M&amp;PW answers:

Page 39-40

and here  

Somewhere in here probably :-).

Related;

Comment in passing bottom of page 1

and

Here