I see now, the real issue is that there are "dangling" connections that need to be cleaned up automatically. In that case, a connection cleanup script is probably the best way to go if you can't figure out why the connections aren't closing. It would enumerate the active connections and terminate the JRA connections with an idle time larger than a predefined age. In RepositoryAccess you would accomplish it like this:

foreach (SessionInfo otherSession in Repository.EnumSessions(session))
{
    if (otherSession.SessionId == session.Id)
        continue; // Current session

    if (otherSession.ApplicationName.Contains("LFJRA"))
    {
        TimeSpan idleTime = DateTime.UtcNow.Subtract(otherSession.LastActivityUtc);
        if (idleTime.TotalMinutes > 60)
            Repository.TerminateSession(otherSession.SessionId, session);
    }
}