Does your enrichment valve close all the way.  When you pull out the enrichment plunger (commonly & incorrectly referred to as the "choke"), it opens up a fuel passage along with a bypass air passage.  That raises idle speed and admits a big shot of fuel.  Does the plunger go all the way home when you push it in?

That plunger has a rubber seat pressed into the face of the plunger.  Is the rubber seat in there?  If the rubber seat is missing or damaged it will allow fuel to leak into the bypass passage and cause a rich mixture.

I forgot to mention your slide needle position.  You must have removed the needle because you stated it has three grooves.  If the needle is assembled incorrectly, it will cause an overly rich or lean mixture.   Tell how your slide needle assembly is stacked up, position of the clip, etc. 

-The clip should be in the center groove.

-The plastic spacer/washer should be on top of the clip.

-The thin metal washer should be under the clip.

-The spring should be under the thin metal washer.

Also, verify that the retaining plate is installed correctly.  The plate should not cover the holes/orifices in the bottom of the slide.  It's possible to install the retaining plate upside down, which will obstruct the holes/orifices.  When it is installed correctly, it should look like this.

Jeremy, I don't recall you saying exactly what slide needle you have in there.  You mentioned you thought it was the stock needle with three grooves.  Can you look at the needle and tell us if it has the Mikuni logo and what the markings are?

The needle jet and slide needle work together to control your mixture.  You spend about 90% of your operating time with those two components in control of your mixture.  They are very precise.  

Mikuni needle jets generally come in various sizes.  They are sized in .0002" increments.  That's two ten-thousndths of an inch (.00508 mm).  For example, changing from a Q-7 to a Q-8 only increases the orifice size by .0002".

The slide needles are equally precise.  The stock slide needle diameter only changes by about .0005" for every millimeter of movement up or down.  Move the needle up 1mm and the diameter in way of the needle jet gets smaller by about .0005".

These are not components that lend themselves to mix & match activity.  Taking parts from an aftermarket kit and mixing them in with another part of unknown origin or quality is a recipe for disaster.  

You can't buy a kit from Suzuki.  The parts are sold individually.  I suggest you purchase genuine parts from a Suzuki dealer or an on-line parts supplier.  Babbitt's Suzuki Partshouse lists the needle jet for a 2003 LS650 as 09494-00861 and the price is $45.49.  They list the slide needle as 13383-24C40 and the price is $16.16.  With shipping charges I suspect the total cost for those two parts will approach $80 bucks.

Genuine Mikuni main jets and pilot jets can be purchased all over the place.  You can get them at a Suzuki dealer, on-line at Jets-R-Us, from most motorcycle dealers, Sudco, independent repair shops, etc.  Again, look for the Mikuni logo to give you some degree of confidence that it is a genuine Mikuni jet.  These days, even the logo is no guarantee, so buying from a reputable outfit is important.

You seem to be in the ballpark with your main jet and pilot jet.  No way those sizes would result in that black sooty spark plug.  The pilot air jets up top (under the diaphragm) should be a #230 and a #45.

What did you do about your spark plug?  Did you switch to the correct NGK plug?  It would be a shame to struggle with your carburetor only to find that the E3 plug was the source of your trouble.  Once you get it running correctly, you can always try the E3 again.  Until you get it running correctly, I wouldn't complicate things by throwing another variable into the mix.