i think you are on the right track.
not sure why you need the florescent tube lights tho (those should be greenish rather than yellow btw)
you should look into using portal lights, they will take colour information from your outside environment and bleed the colour in a kind of indirect diffuse way. area lights can be slow to render tho.

i set this up really quickly. there is no final gather its just two area portal lights, a directional for the sun and a couple of bounce lights. it renders fairly quickly.