Your sentence is grammatically correct. Just simplify it and you will find the very common use of neither as a conjunction with nor. Note: Cambridge Dictionary shows that:
We can use neither as a conjunction with nor. It connects
two or more negative alternatives. This can sound formal in speaking:
Therefore, they neither need X, nor depend on B
So already your initial sentence is formal enough. If you wish to use an inversion, there are two ways you can rephrase it:
Therefore, neither do they need someone else to flatter them, nor (do they) depend in great part on a government pension.
Therefore, they do not need someone else to flatter them, neither do they depend in great part on a government pension.