Cannot open database "dbname" requested by login.The login failed.Login failed for user 'DOMAIN\Windowsusername'
That login probably has the dropped database listed as his/her default database. You just need to assign a different default database.

In SSMS, connect to the server, then go to the Security section, then the Logins section. Find the login in question, right-click, then choose Properties. On the General tab, you'll find a "Default database" field. It probably has the name of the dropped database. Select a database that is still valid, and use the OK button to save your change.