I have a game room on the 2nd level of my home. I need to convert it into a room. How do I go about making the sub wall into a complete wall?

Welcome Mandi,
U can add to the sub wall(knee wall) to bring it to ceiling. If there is a cap(shelf) on top of knee wall, u will have to remove it. Use the same size studs for the new section of wall. If the wall is not to long and u have help, u can frame wall on floor and lift into position. Easier then building in place.

Chuck

Since this isn't a bearing wall, you can do as Chuck says. Stick build a wall from the same size lumber as the existing wall, set it on the knee wall, nail or screw together and the drywall or panel.