I've studied the electronics for the flashers extensively in the past--I was designing an Arduino controller that would, among other things, control the flashers so that, among other things, I could use LED lights.
Long story short, all the circuitry in the K-bike is designed for incandescent bulbs and the type of load that they present. LEDs are diodes, electrically speaking, and present a different kind of load, not to mention a much higher resistance / lower current load. I'm not surprised things went hay-wire. The bulb monitor, for instance, will always think the bulbs are burnt out. The flasher also won't work properly. Unfortunately, the flasher is not a standard automotive type that you can buy at your local store and thus replace by simply plugging in a new one designed for use with LEDs.
The simplest way to solve it while keeping LED lights is to use what's called a "load resistor". It's a resistor (which is basically what an incandescent light bulb is, electrically speaking) that presents the same type of load to the electrical system as an incandescent bulb would. Thus, all the circuitry should work properly. It does, however, draw just as much power as an incandescent bulb (which defeats some of the purpose of using LEDs in the first place), except instead of emitting light, it generates heat. That's why they have heat sinks on them. You'll need one for each bulb. Try to match them with the same wattage as the incandescent bulb being substituted. They hook up in parallel with the LEDs. LEDs draw so little current by comparison that it typically works just fine. I've never known someone who did this with a K-bike, however, so no guarantees...
