One possibility would be to swap the living room and the dining room. Since the current living room is larger than the current dining room you could still keep some comfortable seating (couch and upholstered chairs?) against the front wall for a keeping room/dining concept. Then your new dining areas (kitchen and dining room) and living areas (living and drawing room) would have good flow between them, which would work well both for family and entertaining. You could also add some builtin bookshelves in the drawing room or new living room to create a library area to add another fun element to the living areas. These bookshelves could be against the long walls in the drawing room, around the window with a window seat in the living room (former dining room), or built to create some separation in the open area between the drawing room and living room/stair area. Your plan seems to show some sort of builtin there, but I'm not sure what it is.

So much depends on your lifestyle. Will you typically be eating out in that dining area, or is that just for special occasions, and other meals will you want a convenient/cozy place to eat in the kitchen? Are you preparing food for large or small family? When you entertain, will people gather in the kitchen while food is being prepared, or would you prefer the food prep to be more out of sight? With the kitchen tucked away in the corner like this, think about the flow between the kitchen and out into your dining and living spaces, for how you envision using the space. Whatever is placed in that far corner will be farthest away from the dining/living area, and possibly require a trip around your island, so keep that in mind for convenience consideration.

I think a comfortable distance between the island and the counter is around 42". It seems that you have only 30" between your cook top and the counter. I might be a little too tight. The wall between the entrance and the kitchen could be 4' long instead of 6', you would then be able to move the island a foot which would give you more room between the cook top and the island. Could you switch the location of the toilet and vanity in your dad's bathroom so your enter facing a vanity instead of a toilet which is a lot nicer.

I would consider putting cabinets on the short wall where the utility wall is. Have your stove there centered on the area. Move the sink so it is centered on the long wall where the cooktop is now. Put the fridge on the end run of cabinets closest to the dining table. That will give you an L shape kitchen with a nice work triangle. If you have your heart set on an island remove all the wall cabinets and install a large island. This is not the style of your kitchen but it does show the layout I described above. L shape cabinetry with stove on short end, sink in the middle with DW to right of sink with cabinet above it for dish storage and the refridgerator on the end run so it is out of work triangle and conveniet to the dining area for serving or work triangle for prep and cooking. Island can be any style you desire. L shape with island and open across from island; you might consider this layout if you desire an island. Something like this?