Look at this video at 8:20 to see how the lever goes in and gets positioned correctly.
To remove the broken cable you need to remove the bolt that holds the cable in place just forward of that lever. This is the number two PITA bolt on the KLR (number one being the back bolt on t he cam chain tensioner).
Once that's out, remove the broken cable and route the new cable into place, first putting the drum end of the cable into the handle bar clutch lever..
Then get the clutch actuating lever positioned as you have it in your picture. Rotate if forward and hook the ball of the cable onto the actuating lever clevis.
With that done, insert the lever back into the case as shown in the video at 8:20.
Now, with one hand, hold the cable down in place so that the cable clamp is over the hole in the case. With your other hand, reach over the header pipe and hold the bolt in place so that it is directly over the threaded hole. With your third hand, use your 10mm socket to tighten the bolt. Now you know why it is the number two PITA bolt.
It can be hard the first 100 times you do it, but you'll get the hang of it. Be careful, though, as it is a short bolt and easy to cross thread becasue of the awkward placement.
If it feels tight, back off and start over; you don't want to strip the threads because you'll have to remove the case, throw it across the garage, then retrieve it and put a stud in the case in lieu of a bolt (which ain't a bad idea and might be Norm's 106th mod). No, you may not ask how I came to know this...