In order to connect to a named instance of SQL Server you will need to configure an alias for that instance as described here: http://support.microsoft.com/kb/265808
That will allow a direct connection to that alias from the HelpSpot Windows installer.
Make sure the full text indexes are created as described here:
If they are make sure the full text service is running. In services it's called: "SQL Full-text Filter Daemon Launcher"
Before starting it make sure in properties it's set to start automatically and that under "log on" it's using the local system user. Save any changes and start/restart the service. You may also need to go into the full text indexes and run a full population and/or apply tracked changes.
In extremely rare cases SQL Server can lose seeding on the HS_Request table or another tables, primary key. If this occurs inserts to the database will fail with this error. You can check if this is the case by running the following in management studio:
DBCC CHECKIDENT ('HS_Request');
If the current seed value doesn't match the current column value (it will be less) than there's a problem. You can reseed the tables primary key with this command:
DBCC CHECKIDENT ('HS_Request',RESEED,######);
where ###### is the new ID value. Make this a few higher than the current highest value in the table. That should resolve the issue.