Ok, what is this broken hose (8 inch rear-end)

It's the axle vent. If you unscrew the "bolt" that holds down the brake hose bracket to the axle, you'll actually see it is a fitting that goes into a hole in the axle tube. The hose simply extends to a place away from any splashing water.

If it is too hard and brittle to put it back just replace it.

Sounds like the rear axle vent line. No major problem. If it is plugged, the rear axle cannot vent properly and as heat builds up pressure, the pressure will vent through one of the other seals. This is not good, don't ask how I know. It attaches to a clip on the under side or the car, high above the axle. Also, the point it attaches to the axle housing at can be unscrewed and cleaned. Mine was blocked real tight and I used a small drill bit and a hand (non powered) drill to clean it. Any proper size hose will work unless you are going concours.