I strongly suspect the agents at DTW will kick the can over to PHL to "fix" the problem. They'll say "we show you confirmed in business, you can get your boarding pass at the gate in PHL."
Not sure how long the connection is, but time obviously works against OP. Maybe PHL-MAN is the best option at that point, if there are even any seats left by then. Or maybe they could get OP on something like PHL-AMS-EDI, using KL on the second segment (I haven't checked timing or availability)