You dont need the github client to download the source. You can just download the source as a zip. I guess you have done some stepps, but I wrote down everything from scratch.
Sign Up at Github.
Go to https://www.unrealengine.com/settings
Under Accounts “UPDATE GITHUB ACCOUNT:” type in your Github user name, if you haven’t done already.
Login at github.
Go to https://github.com/EpicGames/UnrealEngine (you have to be logged in)
Use branch 4.4 (or the branch you want to work with)
On the right side you see “Download Zip”, hit it and a download will start.
Scroll a bit down and also download “Required_1of2.zip and Required_2of2.zip.”
Now extract the first Zip and extract the other two into the created folder.
Now you can run “GenerateProjectFiles.bat”, open the UE4.sln and compile the Editor.
Some more links:
https://github.com/EpicGames/UnrealEngine
https://www.youtube.com/watch?v=usjlNHPn-jo&feature=youtu.be
Big Edit:
Haven’t seen that you have already the source code. Sorry for that…