SQL Server Permissions – Fixing Broken NTFS Permissions for MSSQL$SQLEXPRESS

I have been "securing" our database server. To do this I removed some permissions from the folders that the SQL Express install created.

D:\SQL
D:\SQL\Data
D:\SQL\Backup
etc..

I kept the permissions on the Data folder as they were, but I removed anything I thought I didn't need from the others.

Now I can't backup databases. If I try selecting the backup location the dialog doesn't show anything in the d:\SQL folder and if I put the path in myself it tells me this:-

Cannot access the specified path or file on the server. Verify that you have the necessary security privilages and that the path and file exists.

…

It looks like folder permission problem. Another server configured in a similar way has a bunch of "allow" permissions for an account called "MSSQL$SQLEXPRESS", (which I know I deleted), But I can't re-add these permissions because there is no such group, user or built-in security principle on the server.

How do I add permissions for this MSSQL$SQLEXPRESS account?