Edit: There are carb repair kits on eBay for both the newer CV carb and the older slide carb. See post below.
You can buy OEM parts from online vendors, but it helps if you know what parts you really need, instead of buying everything on the parts fische.
ProCylcle has a jet kit, but I don't believe the jet sizes correspond to the stock TK ones.
Your 2006 is now 9 years old. depending on the mileage, it can have worn parts.
If you are not the original owner, there's no telling what has been done to the carb previously.
You said you have cleaned it...but if you didn't soak it and blow out all the internal passageways, it is not really clean.
The most common problem for a bike that has been sitting is a clogged pilot jet/pilot jet circuit.
(I just finished cleaning my 2005 TW carb...again.)