As RappArchitecture mentioned, deciding on the sink/DW and Washer/Dryer locations without looking at the Kitchen layout as a whole severely limits your options. If you are willing to consider other options, here are three.
A couple of important notes:
(1) As you know, you are severely limited in usable storage. I understand you do not want to remove the wall with the built-in. Have you considered, instead, removing the 17" x 5" wall and the 28.5" wall and extending the 42.25" wall another 31.2"? This would give you much more usable space. It keeps the Linen Closet but opens up the Kitchen space. from your drawing it appears that space is a hall that leads to the Bedrooms & Bathroom. If your drawing is accurate, you should be able to make these changes without adversely impacting the overall flow.
From this:
To this:
(2) The washer & dryer -- probably the biggest obstacle to a decent layout. The first two only work if you can find another place for the washer & dryer -- maybe in the Bedroom/Bathroom part of the home (which makes the most sense in general since the vast majority of laundry comes from and returns to the Bedrooms). We'd have to see a full-floor layout to see if there's another place for them. [Ideally, the dryer should be on an exterior wall for exterior venting. My mother has a "ventless" dryer and it takes forever for even small loads of clothing to dry! And, she has to empty a reservoir each time (the reservoir is where the moisture is deposited once it's removed from the clothes).] By moving the W/D elsewhere, you gain a lot of storage space!
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For your consideration:
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Includes an alternate upper cabinet arrangement (which would work in any of the layouts)
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Washer & dryer staying in the Kitchen
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Backsplash
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