This was pointed out as ungrammatical in a process document. Its clumsy, but is it ungrammatical? It is the use of immediately unmediated by something like 'after' which is at issue.
closed as general reference by FumbleFingers, StoneyB, Kristina Lopez, JLG, Andrew Leach Feb 13 at 20:27
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There's a missing preposition before "the last day". By, before, on, after. Something.
Also, there's a semantic (not grammatical) mismatch between specifying a deadline and saying it should be done "immediately".
It is not grammatical, in that the noun phrase “the last day of that member of staff” has been tacked onto the sentence after the adverb immediately, in a position where either nothing or a prepositional phrase would be suitable. Perhaps one of the following forms would work, with choice of form depending on what you mean and what you want to emphasize: