Insert Global Site Search Into Network Theme

hi guys,
I removed the Network Theme's built-in search box, which is typically located here:

But I'd like to insert the Global Site Search plugin in its place instead (to have the search bar neatly placed where the former one used to be.)

I don't know exactly which chunks of code I should borrow from the Global Search plugin php file, and where exactly I should insert it within the theme's "discover-header" file.

Both files are here below, in case anyone wants to take a stab at it or guide me? Feel free to modify; they are just backups.

Thanks so much!