Would you be kind enough to tell me whether the following two sentences are enjoying the same rights?
She did what I desired.
She did that which I desired.
Thank you for your efforts.
Yes and no. The former is less archaic, the latter very so.
Even "desired" is rather dated compared to "what" as a conjunction.
Natural English in our times would prefer: "She did what I wanted."
Or, more formally, "She did as I desired."