We have recently discovered that the shower in the master bedroom
above the kitchen is leaking. We learned about it because the ceiling
in the kitched started showing signs of water.

I had a similar problem and found a $9 solution. Maybe this can help.

I found that the leak came from the junction where the drain met the shower
floor. For about $8 you can buy a tool that you can use to unscrew the drain
from the pipe that goes into the floor. I discovered that the jerk plumber
that installed the shower was very stingy on the plumbers putty and there was a
leak path. I bought some putty (about $1), took out the old putty and put in
an adequate amount. Problem solved.

One thing to note, my leak seemed to only happen when the tub was used. That
makes sense if you think that the water will take the path of least resistance.
When the shower was in use, the water went down the drain. When the tub was
in use, the water seeped around the seal.

Hope this helps.