When you are installing LED turn signals you need to do something to prevent the LEDs from seeing a path to ground through the indicator light. If they see that path then they will have current running through them and will operate. The usual symptom is that when you flip the turn signal switch in either direction both the right and left sides will flash.
You can either install diodes in the indicator circuit (messy) or install a non-polarized 194 LED bulb in place of the incandescent. I haven't looked at all of the 194 bulbs, but I think they are all non-polarized. You can check the specs, like for this one
. Down in the specs where it says "Polarity Sensitive: No
" is what you would be looking for. Note that this bulb may not be the one you want, but be careful as some of the 194 have a bigger head than the hole in the KLRs instrument cluster. You can get those in, but it's a bit of a wiggle.
All of this is independent of the flasher.
Are you doing this on your 2005 or 2012? The LF-S-1-FLT is a plug 'n play for the Gen 2. For the Gen 1 you could get this one
or this one
. I don't see why they would not work but do your research. The bullet type would also work, but you'd need to provide the connection. Don't get the blue one; I've had bad luck with it.
All of the info on the 194 applies to the Gen 2. The Gen 1 takes a #24 bulb and I don't see a non-polarized one. Never did turn signals on a Gen 1 so I'm not sure what the answer is. Is there anything in your kit?