深いお詫びの意を示したいときって英語でなんて言うの？

These are all formal sentences that can be used in a business email.

There are many ways to apologize but when it comes to business emails, we use the above examples. :)

1. ..so, so.. We use this form when wishing to add stress to an adjective.
   "James is so, so strong!"
2. I can't express how + adjective = We use thiis construction to add emphasis to the given adjective. "I can't express how happy I am to hear of the b irth of your new baby!"
3. to be knocked for six = knock somebody for six. (British & Australian informal) to surprise and upset someone a lot
   "It really knocked me for six when my ex-boyfriend announced he was getting married"

Sincere - proceeding from genuine feelings
Regret - feeling sadness or disappointment over something one has done or something that has happened
Wholeheartedly - with complete honesty
_
Examples

I apologize wholeheartedly for failing to let you know that the meeting has been cancelled.
Please accept my sincere apology for the late response.
I apologize wholeheartedly for the inconvenience.

You can use any of these :

"Please accept my apologies."
"Please accept my sincere apologies for…"
"I would like to express my deep regrets for…"
"Please accept my apology for…"
"I’m particularly sorry for…"
"Please allow me to apologize for…"
"Please accept this as my formal apology for…"

If you want to express the fact that you are very sorry for something, especially in a business email, you can say:
"I sincerely apologize."
"Please accept my sincerest apologies."