First of all, for similar phrases like this, I never like to say that they are interchangeable, because that implies that they connotate the same thing. There is usually a best choice to make. While the denotation of the phrases might be interchangeable, providing an incorrect connotation could lead to confusion on the part of the reader or listener of your sentence.
So let discuss connotation with an example.
In doing so:
Bob walked his dog. In doing so, he exercised.
The first phrase in doing so gives the impression that effect of his first action (walking the dog) wasn't at all intended to cause the second. He exercised, but this occurred only as a side benefit.
In this way:
Kathy finished watering her plants and set down the hose. In this way, the hose would be close by for next time.
In this example, the phrase in this way is used. It is implied that Kathy desired the results described in the second sentence. She wants the hose close by. However, it isn't a very strong desire. If her husband comes and stores the hose, she will just get it out again.
So that:
Fred went to Law school so that he could get a degree.
In this example, Fred performed the first action (law school) expressly for the purpose of achieving the second action (get a degree). This was his entire goal.
I'll change out the last example with a different phrase for clarity:
Fred went to Law school. In doing so, he could get a degree.
If you said this, a listener might be confused. They might ask "Did he want the degree?" or "Did he have another reason for going?" Maybe Fred's main purpose was to party with college students and blow off classes.
Conclusion:
Remember that changing out these phrases for each other will still make sense. It will still be grammatical. But it will connotate a different mindset on the part of the subject.
It might help to realize that these are not merely stock phrases, but are productive forms.
The pattern "In [verb]ing, [result]" means that by taking a certain action, a person achieves a particular result. The construction defines the result or the meaning of the action. "In doing so" is a particularly generic example. But it applies to plenty of other verbs. For example, the rules of chess say that if you touch a piece, you have to move that piece. So a person describing these rules could say that "In touching a piece, you force yourself to move that piece next." Or another example from this article: "In making this statement, it is the mother's intent to not comment at this time". This is trying to control the meaning of her action: she states what she is doing, and then the result that she intends for it to have. (Or, in this case, not have.) "In writing a book which is so different from his previous successes, the author is taking a big risk." If it is already understood from context, I could replace "In writing a book which is so different from his previous successes" with "In doing so". (This is just an example of using "doing so" to replace a longer verb phrase; you can interpret this "so" as meaning "that thing I just said".)
The "In [verbing]" constructions frequently take the form of performatives: "In riding this roller coaster, you agree to accept all responsibility for any injuries which you receive." My making that statement affects the legal reality: by defining the meaning of your action, I've turned a basic action into acceptance of a contract.
"In such a way that" is similar to "In [verb]ing", in that it's trying to describe the result of the verb which comes before it. However, this one is more of a stock phrase. It doesn't involve a changing component the way "In [verb]ing" does (with the verb). Here the pattern is something like "[verb phrase] in such a way that [result]". Going back to chess again, an example could be "He moved his knight in such a way that I would lose either my bishop or my queen." Or "He wrote his name in such a way that no one could read it." Some people consider this a less favorable construction than the one above, because it uses more words to mean the same thing. Then again, sometimes people don't want to be concise!
"In a way that" is more general in application: it is not restricted to describing the result of an action, but merely the manner in which it was done. So you could say "He wrote his name in a way that no one could read" (this doesn't specifically indicate the result, it's purely descriptive) or "He whistled in a way that I'd never seen before" (which is absolutely not describing the result of his decision).
Hopefully these will seem more comfortable or more familiar with more practice.
Best Answer
Substituting 'In doing so' for 'In this way' gives pretty much the same meaning as your quoted text. After Kathy finished with the hose, she put it down. The fact that she did that makes it near where she will use it next time.
Substituting 'in such a way that' changes it slightly to put more emphasis on the method Kathy employed to set down the hose. Perhaps she coiled it or moved it out of the way of anything (such as a car) that might be placed on top of it. To make that change, however you would need to join it to the previous sentence ie:
As Tiercelet says, the passage in its current form is somewhat awkward, but not incorrectly so. Substituting 'In doing so' would reduce that awkwardness.