(First off, my comment above is no longer valid since you have altered your sentence a great deal. Your original sentence made little sense.)
The grammar pattern:
「Action A + て + はじめて + Action B」
is a very common one and it would be quite useful if you learned how to use it naturally.
One of the examples from your link, however, is completely irrelevant and it should not be there, I am afraid to say. That example is:
「こんなに腹{はら}が立{た}ったのは生{う}まれて初{はじ}めてだった。」
"It was the first time I felt so angry in my life."
That Japanese sentence itself is perfectly grammatical and natural-sounding, I assure you. The problem, however, is that the sentence does not mean "It was only after I was born that I felt so angry." Hope you can see the problem here.
The 「てはじめて」 that we are discussing here must describe the order of two events. "Only after A happened, B happened."
Your sentence:
「すみません,送ってはじめて間違ってしまったことに気がついた.」
sounds OK. At least every Japanese-speaker would surely understand what you meant to say. (You still need to use the Japanese period, though.)
Careful native speakers/writers, however, would tend to use the phrase 「てはじめて」 only when B is the logical and natural consquence of A. In real life, people often say things like:
「病気をしてはじめて健康のありがたさが分かる。」 "Only after one suffers a desease, does one appreciate the value of good health."
The logic in that example is clear-cut, is it not? It would universally be understood, I reckon. This is why I stated above that your sentence only "sounded OK". It is "alright" even though not everyone would think it is great. At least, there will be no misunderstanding as far as the meaning is concerned.
To improve it, you could say:
「すみません、送った後{あと}にはじめて間違いがあったことに気づきました。」 or
「すみません、間違いがあったことに気づいたのは送った後でした。」
Regarding the first sentence, the insertion of an 「後に」 would produce the same kind of effect as using "only after" instead of just "after". You would be better off if you did not try to translate "only" into a Japanese word and squeeze it into your sentence.