Correct?

這看來是一通便條, 看不出是業務上用還是給朋友的, 建議修改一些地方
Dear xxx
How are you?
---- (解) How do you do 是對話場合的用語, 寫便條通常不會用. 如果是業務來往 / 對方不是朋友, How are you 的問候語也不需要, 直接講要講事就可以, 最多開始寫為 I am pleased to inform you that ....
I would like to inform you that we have sent the shoes to your home address this morning.
----(解) 這句沒有甚麼問題, 但我給home 加上了 address, 因為我覺得如果收件人不是朋友, 單用home語氣太親密了. 如果是朋友, 可以用更casual 的 your place.
Sorry for keeping you waiting so long.
---- (解) sorry for 後面的動詞要名詞化, 加 ing化為動名詞
Have a good day.

What about :
Sorry for having kept you waiting so long .

You don't use How do you do if you already knew this person...
you should use How have you been or How are you, hope you are well.

Can use Sorry for the wait or sorry to keep you waiting. or we apologize for the delay.

You should also put your name with

Best regards,


XYZ

"Sorry to have kept you waiting for so long."

Otherwise the rest is OK.