One broad, soft source of light would be the fastest and easiest to work with; it could light her, and the background at the same time. An umbrella would work, and would be broad in coverage; a small to medium-sized softbox would work; you could also fire a flash unit, or two synch'd together, off of a wall or the ceiling or off of a 3 x 4 foot foamboard panel, or fire the flash through a diffusing panel made of thin white fabric or tracing paper or waxed paper (yes, the kitchen kind!) stretched or taped or glued onto an old window screen frame or a PVC pipe frame (Tinkertools).
In a smaller, low-ceilinged room, a shoot through umbrella will tend to send 60% of the light through the fabric, and the other 40% or so will bounce off, and will hit the walls and ceiling and create what's called ambient spill light. The smaller the room, and the more ambient spill there is, the more it can act as a fill light for shadows, so be aware that a fill light or reflector might not be necessary if you light that set with a shoot-thrugh umbrella.
The patterned vinyl backdrop will tend to photograph a bit darker than white if the shutter speed is fast; slow the shutter down a bit, and it will tick up in brightness; depends on the ISO in use, the f/stop in use, and the shutter speed. At slow dragging the shutter speeds, like from 1/8 to 1/20 second at f/8 at ISO 400, the background will likely photograph about as the human eye sees things.
You could use any number of light modifiers,depending on the desired effect. The Lightsphere is actually quite decent in smaller rooms, where it sends light all over, and gives a broad light that refelcts off of multiple places. If you're on a tight budget, I think the best thing is a piece of foam board angled toward the subject, and a flash aimed right at that board, so it reflects, makes the 1 x 3 inch flash panel (three square inches) into a 30 x 40-inch, 1,200 square inch light source. You can rig up any number of ways to get this $4.99 piece of foam board so it bounces onto the set properly: prop it on a barstool, string, fishing line harness, masking tape, A-clamps, whatever you have on hand.
A 32- to 45-inch umbrella would work well too. Small 28 x 28 inch square softboxes will work too. You can also use a gray fabric backdrop, or a white fabric, or a white wall if you have one available.