If the SQL Server service is running (which based on your description, it is) check the SQL Server error log. You don't need SSMS installed to do this as it is just a text file. File location may vary depending on how it was installed, but Books Online shows the default location here.

The first entry in the log should contain the version/edition information.

If the SQL Server service is running (which based on your description, it is) check the SQL Server error log. You don't need SSMS installed to do this as it is just a text file. File location may vary depending on how it was installed, but Books Online shows the default location here.

The first entry in the log should contain the version/edition information.

Edit: Despite the comment in the question above, this most certainly does tell you the edition:

2014-06-08 00:05:00.050 spid158      Microsoft SQL Server 2012 (SP1) - 11.0.3381.0 (X64) 

Aug 23 2013 20:08:13 

Copyright (c) Microsoft Corporation

Enterprise Edition: Core-based Licensing (64-bit) on Windows NT 6.2 <X64> (Build 9200: )