I think you’ve done a pretty good job of evaluation. If coolant is not leaking on the floor, not getting into the oil, then it likely is going out the tailpipe. That indicates a leaking head gasket. After all, the bike is old and has sat for some time before you started using it again.
I will say that a head gasket failure on the KLR is fairly rare; it's a robust design. Given the bikes age and history, though, it's not impossible.
The general process for evaluating a head is to remove it, use a good straight edge and some feeler gauges to see how much, if any, warp there is, then to have the head skim-milled to make it flat. Any competent machine shop can do all this for you, and can also tell you if the head is salvageable. They can also check for cracks in the head itself. It shouldn’t cost much for the evaluation or the machining.
On the practical side of things, the bike is 15 years old. That means that it’s not worth very much and spending a lot of money isn’t wise. I think you definitely want to go the inexpensive route here. If you remove the head and find it is badly warped, look for a different bike and part this one out.
I do think it is worth the effort to figure out what’s going on and if it is salvageable. If you have decent mechanical skills it is not a really difficult job. If it requires no more than a small skim cut to make the gasket surface flat, go ahead and do that and put it back together.
One thing, though, do not be tempted to pull the head, place a gasket in there, put it all back together and hope it works. A warped head will not allow the gasket to seal for long and doing so is a waste of time and money.
By now you’ve probably scoured the web and found plenty on head gasket replacement, so I won’t dwell on that. You might be interested in going to this post
and looking at the fourth video, starting at about 2minutes in to it, to see what’s involved in installing a head and gasket. You won’t see the teardown, but I think you can figure out what the reverse process would be. The video skips some tedious steps, as it was not intended to cover head installation.
Follow Willy's advice, though. Once you've got the valve cover off, check the valve clearances. If they are tight then that's very likely the reason it won't start. You may find that you can refresh the valve clearance and run it for a bit to see if the water problem is really real. Be cheap!