Login Failed for Domain\ServerName$ when trying to create a new SQL Server Reporting Services Application in SharePoint 2013
Add BOTH Domain\Servername$ login and NT AUTHORITY\Network Service login to SQL Server instance giving you above problem and give both logins the right permissions.

If you are working in a Spanish environment or another language try adding NT AUTHORITY\Servicio de Red  or whatever the equivalent is in your language. 

For steps on adding logins see links: http://support.microsoft.com/kb/889646/en-us  and http://www.c-sharpcorner.com/UploadFile/specialhost/how-to-fix-login-failed-for-user-nt-authority-iusrs-in-share/ and https://stackoverflow.com/questions/2806438/login-failed-for-user-domain-machinename

If you have a SQL Server Instance with logins that have been working for you when accessed by application pools and the like, you might want to check which logins are not in the troublesome SQL Server instance and that you might need.

Other comments:

Ok, even though the error said: Login Failed for user 'Domain\Servername$' and I added that login to the problematic SQL Server instance per what I'd seen described on the internet, it seems that the login I ALSO needed to add was NT AUTHORITY\Network Service.

I basically decided to add any logins that were in my normal SQL Server instance that I was using for the regular SharePoint databases and that weren't in the SQL Server instance I was using for SharePoint Reporting Services to the latter. Something else that tripped me up when I was looking originally to add the NT AUTHORITY\Network Service login was that I couldn't find it. It turns out I had to add it in Spanish as NT AUTHORITY\Servicio de red, which is wierd because the app pool has its Identity in English. I guess this happens when some systems have English as default language and some others have Spanish