For those interested in trying Pokémon: Sacred Gold and Storm Silver, here is a condensed version of the installation checklist:
Inside the extracted files, locate the folder containing the patching utility (usually named xdeltaUI or similar).
Click "Open" next to the Source line. Select your clean, original .nds ROM file.
Click "Select ROM" and choose your clean HeartGold or SoulSilver .nds file.