You're right that the sound blankets will do little if anything to soundproof the room. They mostly just serve to reduce echo.
Hanging MLV curtains on the walls will probably make a noticeable difference at higher frequencies, but won't do much to stop bass. You can get somewhat better results if the curtains are separated from the walls by a good distance (a few feet) but again it's still not going to do a lot to stop bass. You can add in a riser that isolates the sound from the floor, for a bigger improvement.
But your only truly effective option if you can't modify the existing construction is to build a room within a room, with its own isolated floor and ceiling.