I am currently a DDK so I wonder what books helped you to improve your game at this stage.
Of course, there are lots of good go books, however what books do contain theory and techniques specific to this stage?

When I was DDK, Lessons in the Fundamentals was pretty much the only available book for that purpose, but I was already 10 kyu when I read it and it contains only a small part of the necessary theory for reaching SDK. (I got the remainining theory from advice in clubs, especially "learn from your own mistakes".)

If you want to get all the needed advice from books and you can't develop significant parts of it by playing or getting verbal advice, you need books for general theory, life + death, and reading. (I am not sure if Tesuji (Davies), which contains tesuji techniques but otherwise little theory, is already suitable for DDK; I read it as, IIRC, 9k.)

Since you ask for books with theory and written specifically for the purpose of improving from DDK to SDK, I'd like to mention that I have written two such books (although they can still be read as SDK).

For general theory: First Fundamentals explains all relevant DDK mistakes I found in DDK games, except "advanced" life and death or reading mistakes. Presumably others recommend also other books, but their common problem is that by far they do not explain all relevant DDK mistakes; IMO, their authors have not thoroughly studied enough DDK games to find all relevant mistakes. This does not mean that other books are useless, but to get a comprehensive explanation of general theory mistakes, I think the mentioned book is essential.

For life and death: Life and Death Problems 1. I am not convinced of other L+D books for your desired purpose, but am sure there will also be others with other suggestions ranging from problems only via also techniques to also theory and techniques. Mine is, AFAIK, the book for your purpose reaching the farthest to the theory side of study, although it still is mainly a problem book.

For reading: Currently, I cannot recommend a particular book specific on reading, with theory, written for DDK to SDK improvement. Instead, pretty much the best you can do today is still problem books. L+D problems are only part of it; you need also non-L+D problems, and the Graded Go Problems for Beginners series is often recommended (it appeared when I was too strong for it, but from occasionally browsing through it, I think it does indeed serve your purpose, although I cannot tell you with which volume to start).

More topics can be mentioned, but, for DDK to SDK, their importance pales in comparison to the three mentioned topics. Learn about them, if this does not make you SDK yet, find out what of these three topics you still don't fully apply; only then other topics are seriously worth considering. I know that some like to suggest opening, but I say: merry Xmas, all you need to know about the opening are the principles of general theory, such as choosing the big and valuable.