Sorry...I can't really tell what I'm looking at. Is this a window opening? Also, that doesn't look like sheetrock...is it? Finally, even if they told the truth and it hasn't leaked (at least visibly) in years, you could have old or continuing structural damage inside the wall. This will require some investigation on your part....

How to repair????
Just tear out the bad material, replace it then do a nice job finishing it.
My biggest questions would be what caused it and has it been perminatly repaired. That looks like commercial construction judging from wall cavity. Does the building have a flat roof?

That looks like some fine old plaster. Make sure the leak has been stopped then give it a skim coat of mud. I'd sure try that before ripping it out and drywalling.

Trebuchet-
I generally agree with you to not make more work than necessary, but it may be hard to confirm there isn't a leak going on internally without at least some modest demo. The last time I dealt with a wall leak, there was little external evidence of damage, but, inside the wall, the studs had literally been turned to mush...I was able to scoop out handfuls of sopped wood pulp and moisture ants using just my hand.

I would say do nothing until you know with certainty that it no longer leaks there. Has it rained seen you moved in?