幸い、[sentence] 幸いなことに、[sentence] 幸いにも、[sentence]
幸い and 幸いなことに mean "fortunately". They open the sentence, and what follows is something that turned out well when it might have gone badly.
幸い can also come after the first half of a sentence, right before the lucky part, as in the last example.
幸いなことに says the same thing with more weight on the speaker's relief. It belongs to the family of おどろいたことに驚いたことに ("surprisingly") and ざんねん残念なことに ("unfortunately"): a feeling about a fact, put up front.
What follows is a fact that has happened or is the case. A wish, a plan or a request doesn't fit.
These go beyond "fortunately" at the start of a sentence.
At the end of a request in a business email, 幸い means "I would be grateful". It follows a conditional, most often 〜ていただけると.