by doing that we managed to solve the problem

[azz]

a. Nancy came up with the idea of using another medication. Thus, we managed to solve the problem. Jeff's digestion is fine now.
b. Nancy came up with the idea of using another medication. By that we managed to solve the problem. Jeff's digestion is fine now.
c. Nancy came up with the idea of using another medication. With that we managed to solve the problem. Jeff's digestion is fine now.
d. Nancy came up with the idea of using another medication. Through that we managed to solve the problem. Jeff's digestion is fine now.
e. Nancy came up with the idea of using another medication. From that we managed to solve the problem. Jeff's digestion is fine now.

Which are correct?

 

[Tarheel]

I'm not sure about the last one. Otherwise they all look fine to me.

 

[teechar]

The second is also not natural. The first is okay in a formal (or semi-formal) context. Otherwise, you can simplify it by saying, e.g.:
"... medication, and that's how we managed to solve the problem... etc." or "... medication, which helped/enabled us to solve the problem... etc."

 

[SoothingDave]

That second sentence is not really necessary at all. Nancy came up with this idea. Jeff is fine now.