If you suspect its the coil or whip, take it off. Its not used for 450Mhz at all, only for a very narrow section of the 30-50 Mhz band. If its shorted out, removing it should allow the actual discone to work. Then you can put a Ohm meter on the disc and the cone, and see if there is any continuity which there should not be if the coax is unplugged from the scanner. That would narrow down the problem.
Mine is picking up several sites right now, and I have it sitting on a tripod 3 feet off the ground in the back yard, for maintenance (two radials broke off during the ice storm, and Im going to replace them all with solid rods instead of those floppy tubes radio shack used).
But to answer your original question, you can make a 450 Mhz coaxial dipole very easily. It will have about 2dbi gain, and can be shoved up into a piece of PVC if you wanted to permanently mount it. You can use 50 or 75 ohm coax, to match whatever coax you use for a downlead, and radio shack has the connectors for a couple bucks. Whole thing shouldnt cost more than $10.
You basically take a section of coax, measure 1/4 wavelength from the end, and at that point cut off the outer jacket (without cutting the sheild wires) and remove it. Then carefully roll the shield backwards over the jacket leaving you with a 1/4 wave of plastic or foam covered center conductor on top, and shield over the outer jacket on the bottom, Seal that in a pvc pipe, and solder the coax end to a female jack and mount it in a pvc cap at the bottom.