I have a 2001 Park Avenue and the Rear Defroster doesn't work anymore,how can I fix it?

Could be a bad relay !
The rear defogger circuit is protected by a 40A fuse. Voltage is supplied through the fuse to the rear defogger relay. When you press the rear defogger switch, the heater and A/C programmer sends a class 2 message to the body control module (BCM), to ground the rear defogger relay coil output circuit. This in turn closes the relay contacts allowing current to the rear window grid.
Do you know what a class 2 message is ? What a BCM is ?
HTD BK relay in rear fuse box ,location In the center right rear of the passenger compartment, under the rear seat cushion
Watch this video on testing from relay base ( where it plug's into )
Do
VEHICLE RELAYS Testing from the relay base

none, if it lights up then power is going to the defroster. the problem is that you have a small tear in the defroster grid

Take it back to them, don't let them guess what the problem is, hell a shade tree mechanic can do that, and don't pay them anymore money, fix it right. running a wire means there is a broken, shorted, grounded wire, for $500, I would be pissed.

you need to check the fuse on both sides with a test light for the rear defroster

Regarding the repair of the rear window defroster. There is a "electric ACD module" on the inside rear driver side roof sail that connects to the rear defroster and antenna. I had the same problem and found the connector for the defroster had overheated and melted resulting in a failed connection. The NAPA part numbers for the module and replacement connector are 25623688 and 15305949.

The common problem with the rear defrost not working on a park ave is the antenna integration modules on the rear c-pillars. This is the way they make the rear defrost and antenna part of the same grid on the rear window. If your external mirrors are heating up, the rear window is getting juice too. They are fed from the same circuit. And yes, the relay does go in a particular way. Compare it to the position of the others nearby and insert with the same orientation.

It sounds like you don't have a ground going to those areas. I think I would check for a ground...maybe pull the fuse on the rear defroster and see if it is even a part of the problem.

Check for Blown FUSEon both Items , Then Check for BAD RELAYS on BOTH Items.

I just accessed mine. It is under the rear seat. You just lift up on the seat, and it pops right out. If you are having problems with the rear defroster, it could very likely be the control module. This is a combination module which the rear antenna, and the defroster grid plugs in to.
When mine went out I smelled a "burnt wire" smell, then it went out.
To access the module, sit in the back seat, drivers side. There is a trim panel that will be by your left ear as you are looking forward. It is the panel that flares down from the ceiling to the rear deck. Gently pull at the edge of this panel, working around the edges, and the small strip which continues over the top of the door.
The seatbelt goes through this panel, so it will be hanging from the seatbelt at this point. There is a ~3" square, black module with the defroster input and output wires, and the antenna wire plugged into it.
You just unplug the wires, pull the one mounting screw, and voila!
On mine the module was obviously melted, as was the wire going into it. I got the replacement from a salvage yard, as the new one was over $50, and the replacement wire was $18. The salvage yard was unable to look it up, so I had to go out to the yard with them and pull it myself. The final cost was $17 (they tried to charge me $40, and I told them I could get it online for that, and they dropped the price).
You can use a crimp on **** connector to put in the replacement wire, but because of the heat in that area I chose to solder the wires together.
BTW, this module fits a wide range of Buicks, and full size Olds.
Hope this helps.

First place to check is under the wide trim piece that runs over the back drivers side door and also runs to the back window. You can pull this away from the body. It just snaps in place and the seatbelt goes through it. Pull it down toward you and you should see a black box fastened to the side of the window pillar. It should have three wire connections. One is the radio antenna, one goes to the window and one comes from the wire harness. Check the two wire connections from the harness and to the window. See if they are loose or in my case the one from the harness was burnt and had melted the black box where the wire went in. I had to replace the the wire connector and the black box ($72.00) and replaced the connector terminals. The replacement connector is available but that little thing is pricey ($32.00). They must be proud of that thing.
Hope this helped.