I had a similar problem, and as my KLR is new to me I really had no idea where to start. Bought a new battery (fully charged), hooked it up and the bike fired right up no problem. Did so every time for about 5 starts, then nothing.
Thought maybe I'd left a light on or something - trickle charged the battery until full, tried to start the bike, still nothing.
Turns out the starter relay switch had come loose. I have a leaky petcock and so stuffed a shop towel underneath it. This in turn dislodged the starter relay behind it and that prevented the bike from starting.
Worth a shot?