I tried georeferencing it from the corners but results were not looking good. A few things, even in the corners where accuracy should be good, were way out. So I digitised a few more control points from the raster with reference to features on OpenStreetMap and did a thin-plate spline warp to EPSG:4326.
Here's the result over Stamen Toner - note the curved edges which are the warped edges of the raster, and the slight fringing where the overlay of the raster and the OSM Australia are slightly misaligned. Its pretty close, perhaps a little more care around Tasmania might get better results:
I'm not sure if warping like this is a better solution for your problem than trying to work out the CRS parameters and scale of the image from the description, but might work for you.
I have just tried with the following workflow to use the Lambert conformal conic (LCC) projection given above (but with lat_2=-40):
- Create four points at the given corners in lat-long.
- Transform corners to LCC.
- Use transformed corners to compute World file for jpeg (pixel size, top left corner) in LCC coordinates.
- Load JPG into QGIS, set CRS to the LCC one, it gets its location from the World file