The Server Name in your first screenshot shows GSMIDBLPRD, but the Instance value in your second screenshot shows GSMIDBPRD (missing the "L"). Which is correct? Also, what's the name of the actual SQL Server instance? I am guessing it is MSSQLSERVER.
For the ArcGIS connection properties window, try inputting the following syntax:
HOSTNAME\INSTANCENAME which translates to GSMIDBLPRD\MSSQLSERVER or GSMIDBPRD\MSSQLSERVER depending on which hostname spelling is correct.
Also, it may not matter, but I am seeing a different username for both screenshots: prd vs iprd.