Jazz for someone who doesn't like jazz.

george, as you will learn, jazz is a very broad and deep subject.  There are many periods, styles, and sub-sets.  So many that it can be difficult to make recommendations to anyone starting out.  Just look at the variety suggested here so far.

Two of the largest selling jazz albums of all times have been mentioned, Miles' "Kind of Blue" and Brubeck's "Time Out".  I think Brubeck is enjoyed by a broader selection of listeners and as such is more accessible.  Miles is deeper and may required more from the listener to appreciate.

Something like the Smithsonian album can be helpful since it presents a broad perspective time wise and with many styles.

Someone really basic would be Louis "Satchmo" Armstrong, AKA "pops".  He recorded from the late '20s to early '70s.  Here's one link to a classic performance with Jack Teagarden and All Stars -https://www.youtube.com/watch?v=gJFgKfuy1oM
 
Otherwise, I think a pop-oriented selection of jazz may provide one of the easier entries.  One subset is called light jazz.  It is quite popular but to me sounds like what I'd hear waiting in the dentist's office.  I think you can find some real jazz that is not heavy, jarring, or discordant that you can enjoy while offering more substance than the typical light jazz.  Here's the Ramsey Lewis Trio with a song very popular at the time -
https://www.youtube.com/watch?v=jsFST-7Hx-Y&list=RDjsFST-7Hx-Y&start_radio=1&t=3

Another of that type would be the Crusaders.  They started out as the Jazz Crusaders but changed their name in an attempt to appeal to a broader market (few jazz musicians sell albums or concerts in big numbers).  None the less, they were comprised by very good musicians -
https://www.youtube.com/watch?v=OahE2GGaGq4

Lastly to suggest one less pop-oriented example which is still melodic, try this by Bill Evans (the earlier pianist, not to be confused with the later sax player) https://www.youtube.com/watch?v=Nv2GgV34qIg

There is a wide, wonderful world out there in jazz, diverse enough that I believe most anyone can find something they like.  These are only a few suggestions.  Enjoy the hunt!

n80, as I suggested earlier there are a multitude of styles/types of jazz.  That is what makes suggestions to a "newbie" so difficult, and the subject so baffling to the newbie themselves.  Many, many great suggestions have been made here and sampling those can be as good an introduction as any.

After my post I considered suggesting the Ken Burns series, but with so much music to listen to I wasn't sure if you'd be willing to watch that much video?  If you watch more you'll certainly find more to explore.

Contrary to many others here I don't consider KOB a good "introduction" album for someone starting out in jazz.  It is one of the best ever recorded but is subtle and complex enough it will not be appreciated by many newbies.  To your credit you've stated you like it.  That suggests you can skip past many of the more "lightweight" suggestions.

So all I'll offer at this point is based on what you've sampled that you do like, continue exploring other recordings by those artists.  You are bound to run into some dead ends but that's OK, just try more recordings.  With that you'll find side men/women with your preferred artists who are interesting and that will lead you to their recordings.  And so on and so on.

A friend of mine was a bit Torme fan.  He thought this was Mel's best album -

https://www.youtube.com/watch?v=7EqFrGcVJW0

He also considered Torme a better singer than Sinatra.  I stayed away from that debate! ;^)