why is this so difficult?
look at holes in the broken and new hangers - are they identical? i.e. same diameter, threaded or not, countersunk (recessed)?
if not, then you have problem.
you say you are using the old screws and now they do not screw into the frame - how can that be?
is the frame or hanger threaded? i.e. should the screws go into the frame from behind and screw into the hanger?
sorry, it is quite hard to work out from pics and not actually being there.
ah, just seen this:
however the frame doesn't have threaded holes - they also don't sit flush on the old dropout either
looks like the screws go through the fame and screw into the hanger - try fitting them to the frame from th inside i.e. the heads are nearest to the gears and then they screw into the hanger - just a guess.