Don't run out and spend money on a load of new stuff without thinking about how you use the room. So I have some questions. Is this a rented property or do you own it? Is this your main sitting room or is it a secondary snug / playroom? What do you do in this room apart from watch TV? What's your budget? Can you give us a plan - I'm not sure if the door is nearer the window or the sofa
I think your quickest fix would be to sort out the layout, having the TV in front of the window is just depressing, I agree with the others that say moving the sofa to under the window makes sense. Then have the TV on either the wall opposite the window or on the door wall. This gives you the chance to surround the TV with additional storage, maybe cupboards below to store toys and either shelves or pictures around it. This will make the TV less dominant in the room. I'd paint the whole room either a dark colour if it's exclusively a TV snug or a lighter colour if it's your main sitting room (don't do a painted feature wall, they are dated and pointless).
Agree a large rug that covers most of the floor will warm things up. You can get plastic washable rugs but I'd look for a second hand wool one while you've got small DC.
I think the blinds are fine and would not recommend putting stunt curtains in there, that would be unnecessarily fussy. If at a later date you want to change the blind then find some gorgeous fabric and have roman blinds with interlining, they are cozy and practical.
I like your new standard lamp, would change the bulb and shade of the ceiling light and add more sources of light, either table lamps or wall lights (you can get battery ones if short of plugs/don't want to sort out wiring). Lots of little pools of light are cozier than a central light.
Finally, it takes time to make a room you love. Don't rush out and buy a job lot of crap from Dunelm / IKEA / M&S. Think about what is meaningful to you and try and buy second hand / vintage / antique where you can. It will add depth and history to your home.