I don't mind the current colours although a greys and beiges tend to suck light out of a room, a brighter colour rather than just white will help. I agree with much of what Jill says above, with a few tweaks ... the lighting needs attention. The room needs some interest, colour and texture to create depth. Get a larger rug, more colourful, with organic pattern which draws the eye outward. Big enough to sit the furniture on. Pick up some of the colours from the new rug and get some cushions. A couple of plants would help with texture and add life to the room. I know you are using the mirrors to add light and space but the one under the stairs is just reflecting the underneath of the stairs ... try a colourful print of a pathway between trees for example, to draw the eye beyond the room otherwise it's a bit bland. The TV is just on a stand that is too small and a bit low, go for a wider piece or try positioning the unit against the other wall in that same corner so it is facing the sofas, then the TV won't look so out of balance. If you make the room a bit livelier and interesting the light issue won't be such a problem. Consider pale turquoise walls for example rather than just white and add some artwork.

Lots of companies out there do lighting specifically for high ceilings or long drop stairwells e.g: https://www.lightingcompany.co.uk/ceiling-lights-c3/lighting-for-barns-and-high-ceilings-c20, https://www.bespokelights.co.uk/shop-by-room-c59/lighting-for-stairwells-c68 http://www.mailorderlighting.co.uk/lights-for-high-ceilings-t167

Will a drum shade suit this lamp base or would I be better sticking to this shape of shade? If I stick with this, could I use a drum shape on a standard lamp nearby or should I match the shape of the shades?

Both look lovely but always nice adding a bit of colour