It sounds like you have one step down to the living room, which would be approximately the distance of a 2x8 (7 1/4 inches)
I would place the new 2x8 floor joists directly over the existing 2x10 joists running in the same direction. I would use a construction adhesive such as PL400 where the new joists meet the old floor, and fasten down with alternating diagional screws approximately every 12 inches. Use diagional blocking to brace the new joist system. I think this would be the easiest to do and give you the strongest floor.