If singing to a karaoke machine's tinny soundtrack isn't your cup of tea, how about having a live band strum along to your song of choice? Karat offers that option (along with tablet-style gizmos to provide the lyrics to hundreds of songs in English).
Perhaps don't get too carried away – after 10pm the charges go up to ¥2000 per hour and there's a 12% service charge plus 8% tax on top of everything else.