I think I would start with a pretty rug to help define the spot. I see two instruments, perhaps you can add a couple of comfy chairs and use the area as a music space and a baby play area. Do you have another place in the house for the farm table? If it's not too wide it could be an impressive sideboard or a big desk in another spot.

The decor in the other rooms looks great, I'm sure you'll be able to find a solution.

You really need to decide how you would use the space. You don't need it as a dining area, and you already have two lounge areas. I would set it up as a play area and music room. Get a large low pile rug that is comfortable for the baby, and will help define the space. I'd put in one or two comfortable armchairs to use for playing the guitars or reading to the baby. Get a long, low storage system for the toys so they are easy to put away and take out as needed. It's nice to have a space for kids next to the kitchen where you can watch them while working. Sell the table and extra pieces you don't need.

Inspiration

Great ideas! So far I'm thinking it should really be a play area for the baby. It will get the most use that way and she needs more safe areas to play. I really like those storage cabinets with the bench seat in the middle. That picture of the fan you posted we already have that on our family room. Maybe get a matching one to tie it together. I think it would be nice to get a large thick soft rug for her or mayber go nuts with those foam play mats. Probably just need to get rid of that big table altogether. Probably just use a couple comfy chairs instead of a couch to keep the space more open.

Those foam mats off-gas all sorts of bad stuff, I say get a thick wool rug and do a cushy rug pad underneath. It’s just as functional, but so much prettier (and healthier!). Wool is pretty resistant to stains as well.

Good idea of regarding Mats versus the rug. I think a rug would look much nicer.

They are terazzo floors all throughout the house which I redid when we moved in.

That door goes out to the garage so I wouldn't gain any additional light changing the door. The rest of the house is almost all windows and that is the only room that is a little dark. There are several recessed lights which will provide plenty of artificial light.

I attached a couple pictures of the entryway(for fun). At the end of the hallway and next to the kitchen is the room of discussion.

Below we've included a few ideas on what you might be able to do with that space. Please note the furniture and accessories used.

Modern Ranch · More Info

Pacific Palisades Family Room · More Info

Casual Classic Remodel · More Info

so glad you refinished those floors. can't believe the people who actually rip those out to put down laminate! I'd love to have terrazzo.

instead of the plastic mats, you can get a large wool rug (the wool is cushier to sit on) or, visit a carpet store and ask to see their remnants in the back.

you can find a 10'-13' width and have them cut it to whatever length you need. They'll also bind the edges for you so you have a rug that's made of actual carpet.

you could even get the padding glued to the back of it if you want.

Or, if you have a big carpet warehouse near you, go look at the giant rolls they have. I got a nice leopard area rug made this way. super plushy too.

Large area rugs are expensive in wool, but if you do the carpet remnant, you can prob get one done for less than 400 or so.

If you want the play area, then take a look online (pinterest) for playroom ideas. there are thousands. even Ikea 'hacks' show how to do those cabinets w/the sitting areas.

I like the idea for a carpet remnant for a rug!

Not a thick rug or carpet remnant. A thinner, low pile rug is much better for playing when they start using blocks, play houses, games, etc.

I bought a rug pad for our other large area rug in the main living room. Wow! That is so much nicer for the baby, and for me. Thank you for that recommendation! It really changed the comfort level in that room

Ooh so glad you like it! It helps the rug last longer l, as well as feeling so much better on your feet or bum when your you’re down playing with the kids.