Documents sent through Amazon's Personal Document Delivery Service are now automatically archived in your Kindle Library (you can control this from the Content Library page in your Amazon account). So even if you no longer can press the Send to Kindle button in NetGalley, you should have access to the file in your Kindle account.
Go to your Content Library page in your Amazon account.
Then select Docs and if the book appears there after you search for it, click on Deliver to Device.
From the list that appears, select the device you want it sent to.
Then click on Make Changes and your device will sync wirelessly and the book will show up on your Kindle device or app after a short delay.