If you are comfortable to remove the door panel to access the underside of that area, the job becomes DIY. You can purchase panel tools inexpensively online or at local stores. Even Walmart carries them, although I didn't check for local stock.
You can expect to find plastic barbed "rivet" type parts holding the door panel to the metal frame of the door. Once you've popped enough free, you may be able to access the mirror control without removing the entire panel. Use caution to avoid pulling control wiring in such a way as to damage it.
You'll also find that all of the wiring has connectors that can be separated to allow panel removal. Press the release tab on the connector when pulling and avoid pulling on the wires. Apply force to the connector directly.
Once you have the panel off, you can see how the controller is attached. The photos on eBay show a pair of tabs with holes that might be bolt or screw locations.
eBay has used controllers from ten to twenty dollars, not particularly expensive.
If you have a shop perform this task, it will be much more expensive, as the door removal can be time consuming and they will use new parts at new part prices. Consider that a body shop is probably going to be more adept at panel removal and replacement.
To provide additional confidence for the DIY aspect, pop into Google and use "door panel removal 2006 Hyundai Coupe." Many YouTube results to provide some insight. One of them is the window motor replacement which starts with the panel removal.
If you know which end of the screwdriver draws blood, you probably can perform this task.