How do I re-attatch the plug for my rear window defogger to a 2007 Chevy Trailblazer?

The piece that is on the glass is glued on with an ultra violet cured adhesive, or a two part epoxy adhesive. The last one I did was cured by the sun in 24 hours (mix, set and cure).

https://www.youtube.com/watch?v=8IBp0Pkoa24

Gluing it back on

https://www.google.com/search?q=fixing+rear+view+mirror+mounting+plate

Permatex® Rear-view Mirror Repair Kit - Canadian Tire

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Rearview Mirror Repair Kit - Princess Auto

https://www.princessauto.com > product
Home · Auto Repair · Chemicals, Adhesives and Lubricants · Adhesives and Sealants · Instant Adhesives; Rearview Mirror Repair Kit.

Doesn't use a relay , controlled by lift gate control module . Your best bet , take it to a qualified repair shop .
Rear Window Defogger System Components
The rear window defogger system consist of the following components:
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HVAC control module

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Liftgate module (LGM)/endgate module (EGM)

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Rear window defogger grid

Rear Window Defogger Operation (With LGM)
Battery positive voltage is supplied through the LGM 2 fuse to the LGM for the rear window defogger system. When the rear window defogger switch is depressed, the HVAC control module illuminates the rear window defogger indicator, and sends a class 2 message to the LGM to enable the rear window defogger. The LGM enables the rear window defogger by allowing battery positive voltage to the rear window defogger.
When the rear window defogger switch is pressed for the first time, the defogger cycle lasts 10 minutes. Further operation results in 10 minute defogger cycles.
DTC B0283 Electric Rear Defrost Circuit - do you know what DTC'S - diagnostic trouble code's are ?
Circuit Description
The liftgate module (LGM)/endgate module (EGM) monitors the voltage level on the supply voltage circuit of the rear window defogger. The LGM/EGM test the supply voltage circuit of the rear window defogger every 160 milliseconds. With the rear window defogger inactive, the supply voltage circuit of the rear window defogger will have no voltage present. With the rear window defogger active, the supply voltage circuit of the rear window defogger will be near system voltage.

There is a module that controls the rear window defogger grid . LGM/ EGM Liftgate or endgate module .Receive's message on class 2 serial data network from the HVAC control module to turn it on when button pressed . Fuse # 3 30 amp located in the rear fuse box . Do you know where the rear fuse box is ?
Rear Window Defogger Description and Operation
Rear Window Defogger System Components
The rear window defogger system consist of the following components:
• HVAC control module
• Liftgate module (LGM)/endgate module (EGM)
• Rear window defogger grid
Rear Window Defogger Operation (With LGM)
Battery positive voltage is supplied through the LGM 2 fuse to the LGM for the rear window defogger system. When the rear window defogger switch is depressed, the HVAC control module illuminates the rear window defogger indicator, and sends a class 2 message to the LGM to enable the rear window defogger. The LGM enables the rear window defogger by allowing battery positive voltage to the rear window defogger.
When the rear window defogger switch is pressed for the first time, the defogger cycle lasts 10 minutes. Further operation results in 10 minute defogger cycles.
Rear Window Defogger Operation (With EGM)
Battery positive voltage is supplied through the EGM 2 fuse to the EGM for the rear window defogger system. When the rear window defogger switch is depressed, the HVAC control module illuminates the rear window defogger indicator, and sends a class 2 message to the EGM to enable the rear window defogger. The EGM enables the rear window defogger by allowing battery positive voltage to the rear window defogger.
The following conditions must be met for rear window defogger to operate:
• The end gate glass position is not below the index position.
• The window express down movement is not requested.
• The window down movement is not requested.
If the end-gate glass is not fully closed, the DIC displays "Tailgate Glass Down" when the rear defogger operation is attempted.
When the rear window defogger switch is pressed for the first time, the defogger cycle lasts 10 minutes. Further operation results in 10 minute defogger cycles.

Fuse Block - Rear
Beneath the left rear seat

There are two fuse boxes, one is under the hood on drivers side and the other is under second row passenger seat on drivers side.

The rear defogger is controlled by the Electronic Time and Alarm Control Module (ETACM). It receives a signal from the switch, and if the car is running, turns on the rear window defogger. Most frequently the problem is a blown 30A defogger fuse. Other common problems are the defogger switch itself or the switch not being plugged in properly, but there can also be a bad defogger relay, defective ETACM, or wiring issue. If you're capable of doing some electrical diagnosis, I'd encourage you to register for a free account at www.hmaservice.com (requires Internet Explorer and Adobe SVG Viewer) and print the schematic from the site. Also, does the indicator lamp come on when you depress the defogger button,I hope this help you in the right direction now you know how it works

Yes, PERMATEX makes an adhesive to glue them back on. See any good parts store. Part number. 15067

Some auto parts stores sell a adhesive that has copper in it. made by "Permatex" just for that problem. Ask the parts guy.

check for power at the element on the window.

If its broken then it is advantageous to replace it.

You can find Loctite Rear Window Defogger Tab Adhesive, at most major auto parts stores.
It's about a $10 repair kit. The instructions are in the pack.