What else do I need to copy over/edit to make my game appear in the Steam Library?

As you found, the games themselves appear within the steamapps\common folder. But in order for Steam to know about the game, you also need the related appmanifest file. These are found within steamapps folder. Example appmanifest_440.acf is necessary for Team Fortress 2.

On a related note, if you need to check where else Steam thinks games are installed, within the Steam client navigate to Steam>Settings>Downloads>Steam Library Folders.

So even though this has a completely reasonable answer already...

I had a similar problem where I'd made a backup of my .\SteamApps\common\ folder and reinstalled Steam, expecting that it would detect the ~1000 or so install directories - but because I'd deleted the app manifests in .\SteamApps\ it was showing that nothing was installed.

I could manually go through, click Install on each game, let it detect that the install files were already there - but that involved a lot of clicking, waiting and trying to figure out what games I'd previously installed.

So instead of doing that, I whipped up a Powershell script to do it for me. In a nutshell, it enumerates the folders in .\SteamApps\common\ and matches them up to app Names and IDs. It spits out a list for you to validate, and then creates an app manifest for each discovered game.

https://github.com/ChiefIntegrator/Steam-GetOnTop