...sounds like there is crud in the carb. When you don't ride a bike for long periods, the deposits of oxidizing fuel end up in the bottom of the carb. My KLR is a '96 that I bought two months ago and I just did a jet upgrade on the carb. I was amazed at how much "crud", that is a scientific term ;-) was in the bowl! You are probably going to man-up and decide to pull the carb out of the bike and rebuild it. Sorry to break the bad news but "simple" fixes are likely to not work. You need to get a carb rebuild kit, because once it's out of the intake and engine side rubbers, you might as well do it right. I think it's your idle jet that is plugged but you can't clean that out w/o removing the carb. Maybe time for a jet upgrade at the same time and then you will be happy for the hassle of yankng the carb out. Keep in mind yanking a single carb out vs. an inline four carb bank is WAY EASIER (don't ask me why I know this!).
Good luck, ask for carb upgrades that cost upwards of $0.22 on this list ;-) and enjoy,
Wrenching on your KLR is half the fun,
Farkled probably WAY too much, but good 20' footer for sure. '96 KLR
, '06 HD VROD, (girl friend hauler
), '05 HD Electra-Glide, heh it's a HD and I'm old now ;-) and I'm proud of riding a 800+ pound bike at my age :-)