sagging floor - attach LVL?
Are you suggesting that you would cut the current 2x6 and then add the lvl to the middle of the span and reattach the 2x6's to the lvl?
I am in a similar situation it sounds like. I am finishing at attic where the floors (ceiling in the living room below) are only 2x6 and the span is 15-20 ft. I get a lot of bounce....not any noticeable permant sag, but definately springy. I am going to place an LVL in the middle of the span. The LVL (actually 2 lvl sandwiched together), will span about 15 feet. I will drywall and finish the lvl so it looks better against my current ceiling.
If you do use an lvl to support below, make sure you have good support for this beam, ie. 2 or 3 2x4 studs together.