I'd like to raise the ceiling height on my second floor but the roof is flat. How do I do this?
Note that this is just a best possible guess based on the current info and in no way should be used as a "given".

Since room is an addition and the fact that you are on the top floor of a flat roof house, I doubt that there is too much to do as far as the roof.  It is quite possible if the room is truly an addition that the roof section could be taken apart and put back up 5 feet higher.  This is all given that your room was really an addition (not just half your room which it sounds like from your description).

Nobody that I know, no engineer I know nor any inspector OK adding on to the existing outer walls - let alone the interior walls.  You have to rebuild.  Taking off the top plate, pulling the electric, taking out windows, and reframing with longer lumber is a quick 1-2 day job for a small crew of 4.  

So you have demo, you have a lot of trash, you have taking apart roof, you have materials (lumber, new roofing, new gutters, new flashing for old roof, new siding, new insulation).  This is all given that you keep the exact same windows, don't move electric or plumbing or hvac, and really don't care that much that this height extension will make the house look awkward since doing aesthetics will add on cost.  I have to say on the really really low end if the roof materials and windows can be saved and no surprises surface and you use low-end builder grade finishes someone might quote you 6-8k.  I would think most estimates will come in closer to the 10-15k range.  Again this is given that this room was a true addition.