Symptoms
Cannot open K2 Designer (Login Failed) with error:
Cannot open database "[DATABASE NAME]" requested by the login. The login failed.
Login failed for user '[SERVICE ACCOUNT]'.
Diagnoses
Users are given the required permissions to the K2 database as per reference in http://help.k2.com/kb001356_Scenario4.
The DB was recently moved from SQL Server 2008 to SQL Server 2014. In the smartforms setup manager, the SQL server was pointing to the old server.
Resolution
Re-run the k2 smartforms setup manager to point to the new SQL server.