Steve, is it possible to identify the trailer.... you know... either from a badge on the frame or by taking a look at the registration? If you are successful with that, your next step would be to contact the facility you intend to purchase the new axle from, ask if they have a data base of trailers.
Quite frankly, I don't think you'd even have to do that if you can use some deductive reasoning. Instead of trying to measure the broken axle itself, why don't you measure the distance from the axle's frame attachment point, double that, and add it to the distance BETWEEN the frame attachment points. A= attachment point on frame to axle end, B = Distance between left and right frame attachment points..... 2A + B = C (where C is total axle length)
easy peasy.
