I am looking for a service, person, or school, that can train me to be
a Techno / Trance nightclub DJ within Chicago.  As a note I am a
beginner.

A list of 3-5 would be excellent.

I have already discovered the Music Industry Workshop (aka DJ Academy)
http://www.miworkshop.com/
and that was a dead end.

Most of these places are dead ends. Trance and Techno are two of the
easiest genres of electronic music to spin. If you were spinning drum
& bass, then I'd understand your desire for assistance. Assuming you
have some rhythm and a few good records, you could really teach
yourself just about everything you'd learn in these classes with one
weekend of good practicing. I can't stress enough how much of a waste
these classes and books are. Get two Technics SL1200 MKII turntables
and some decent cartridges, and a mixer (Pioneer DJM-600 is amazing,
but too expensive for a beginner) and some quality headphones and
start fooling around. Also understand that Trance/Techno DJs in major
metro areas are a dime a dozen and most have been at it a lot longer
than you, so you may want to specialize instead(ie. a less trendy
genre).

Thanks for your comments & specific recommendations.  I may pursue
that route.  My motivation is coming from the fact that I am a working
professional in a completely different career and don't have a ton of
free time at the moment.  I would like to get up to speed as quickly
as possible but this will be a hobby for me - I don't plan to pursue
it as a career.

Perhaps someone can recommend some good websites to learn the ropes
and an affordable mixer for a beginner.

Thanks in advance