Looks like I'm late to the thread. If it's not too late, I strongly recommend you cut a small hole in the ceiling next to the cabinets nearest the apparent leak source. Make it just big enough to get your head and shoulders through, say from one joist to the next and 2 feet long. The point where the water comes through may not be as close as you think to where the problem is. You may need to remove different cabinets than you think, or you may also see what is going on and be able to fix it without removing any of them. Then there is always option C, where you say something regrettable in front of the kids and call a pro.
Seeing exactly what's going on is critical to your planning.