I pulled out some graph paper and tried to see what I could come up with for a 10' x 12' room. (I needed a distraction anyhoo.
) I wasn't exactly sure of how big a few things were (widths for trim, den door, built-in, sink) so my notions may be off.
The more I look at that room, the more convinced I am that the fridge was where the heat vent is. If they used to have a monitor fridge, even though it seems odd to have a fridge right over the heat vent, it would be the most likely place. Also, the presence of the outlet there further leads me to believe that the placement of the fridge for the long haul was right there. They may have raised a later model replacement high enough to allow the heater to still blow forward, but I suspect one of the reasons it's beat up like that was the fridge getting pressed against it. They may have even shut that exchange off.
Based on what folks have posted, I came up with three notions. I checked on some measurements for standards, and also worked from the Baker bungalow kitchen too. (Which, by the way, is only 7' x 11'--that includes the built in cabinet/sink.) It has two feet deep counters, so I worked from that for the window wall except for the mud door corner, where I did a curve there to vanity depth. I assumed sink size, with a single 30" wide since I couldn't see enough of the upstairs kitchen sink to tell if it was double or single.
1) If you take Phil's suggestion and move the toe heater to the sink, that will make it easier to put a fridge there as it will stick out less and won't make the fridge work harder, combating the heater. I checked dimensions, and if you're not married to the idea of a side by side doored fridge, there are plenty of fridges--retro or not--that can fit there, although many will overlap the trim. I have no idea what your budget for the fridge is, but following oak's suggestion, here's another source for slim fridges, many of which are taller as well:
http://www.thekitchn.com/small-and-cool-small-apartment-110535. You can get cheaper fridges with that narrower footprint, but some buyers may be turned off by the overall size, as they are a more standard height. Although there are benefits to keeping the fridge there, I think you would rather have a wider fridge, and tighter work triangle.
2) I know you're leery of making a permanent island, but consider a dishwasher as a smaller islet. If you want to extend the top longways, I'd round the edges of whatever countertop you go with so parents won't worry about kids and corners. Seating or cabinetry can go under the extensions, possibly even flanked with trash/recycling bins, flour door style, or a possible slide out when door opens style (my dad did the latter for trash on one end of our "breakfast counter" abutting the fridge, putting it in easy distance between sink and stove). I used a 30" wide fridge for the L backing the dining room, which is common unless you aim for the bigger side by side models. You can keep the shorter over cabinets on that wall to keep the line, though you may want a deeper model. There is some wiggle room for a wider fridge, but at cost to the corner cabinetry, and a possible can't open the fridge and oven at the same time issue.
Think of the space between your islet and the built ins as a mini pantry. Our pantry here has only a 32" width walkway (measured from drawer pulls on either side), which for most folks is adequate. If I'm estimating right from the picture, it should be fine for opening the lower built in cabinet doors. (It's also a good excuse to steal a smooch when working near to one another on that side of the room, and with all those appliance plugs, I can see a lot of baking there.)
If you go only that far from built in to islet, it should still give you a comfortable passage between window cabinets and floater/islet. It will also give you enough room to open the dishwasher door all the way without banging into any cabinetry based on the models I looked at. You can lengthen it further, I was just trying to see what a smaller footprint would yeild.
I think I read that you're swapping out the sink anyway, so if you bump that towards the left where the dishwasher was, you can put the stove on the window wall, and the fridge near the dining room door. You should still have room enough that if someone is standing at the sink, and another person enters through the mudroom, the person at the sink will be clear of the mudroom door. I was surprised to find that there are scores of folks who are deliberately putting stoves in front of windows now--I thought it would be a code no-no. You also still have room above the window trim for the hood, in place of the short upper cabinet.
The one possible hiccup with this variant is the light switch by the dining room door. If that is a kitchen switch, the fridge may need to be scooched over if you don't want to move the switch to the other side of the door.
The islet variant was the last I worked out on my rough sketch, so I figured it might help if you could see it, so I scanned it. Out of the three ideas, this turned out to be my favorite. Scale is 1 square = 6", and this was a rough WIP, so nothing is perfectly scaled. I put the islet where I did in case a future owner wants to put a table or baker's rack between the bath/mud doors. I also dashed in door swings I knew dimensions for to get an idea of clearance.
3) Lastly, if you feel you simply must remove some of the built in, I'd take that left most side and reinstall it (with needed framing) in the space between the bath/mud doors. It would help keep a closer work triangle, although the layout wouldn't be as optimal distance wise. If you keep the heater there, you can either leave it in place, or bump it forward just enough to make it a toe kick.
I hope the bird's eye view in the picture helps, no matter what you choose.
Classic and timeless. My favorite look for a vintage kitchen and I love black countertops too. We are considering soapstone when we do our kitchen.
It was not because we were friends, this was the pricing he would have bid on any job. My long winded point is - it doesn't cost anything to get an estimate for custom built. 
you could check a local cabinet shop and see what they can get, or a hardwood supplier to see samples. How about letting a cabinet shop glue it up, cut the sink hole, thickness plane and sand it to perfection. I think that would really look special, and you could have real solid wood, not veneer. You wouldn't get the undercounter sink but imagine how the ribbon could stand out with some nice overhead lighting. do a tung oil finish and that could be renewed. To me that would look like a custom kitchen with some class. 
