For family pictures, 5D2 or 6D(if you're willing to wait) might be enough. That way, you can afford a more flexible lens (24-105 or 24-70 mark 1) and a flash with some diffuser. I think having better lenses is the better way of doing it. You won't be getting your nice DOF with just 40mm. 40mm for an FF is too short to get some DOF.
Note: You can also retain your T2i or upgrade to 7D/60D and get a 50mm F1.4 or 85 F1.8 or their faster versions to get that nicer DOF than your present equipment.
I already have the 85 f1.8 and 70-200 f4. My main concern is losing the wide end, the 40 f2.8 would fit that role. Another advantage of going FF would be that getting the 135 f2 for portraits would actually make sense, as right now with a crop it would just be too long for what I would want.