How can I locate and repair a leak on a flat roof?
Clearly this has been a problem for a while and also the standing water and patch job on the flat section of the roof is to blame.
There is no real shortcut here, as you can see, so the solution is to remove the roofing, repair the water damage, and then have it re-roofed properly.  There are ways to apply a flat roof so that it will not leak but slathering it with roof patch is not one of them.