Here is my homework: 

To: Judy White
Subject: Apologizing [1]

Dear Ms. White,

I'm writing to you to apologize for the mistake that we made. The Dispatch Manager has already solved the problems and delivered the a new dress of the right size to you. Could you please send the wrong dress back? Our company will afford cover [2] the [3] expenses.

Please let me know if you have any further queries.

Best Regards., 

Spencer

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[1] 'Apologizing' is a verb, so it does not work well on its own as an email title.  Slightly better wording would be "We Apologize".  However, in dealing with a complaint, it's better to focus on its resolution than on an apology.  A better email title would be "Your new dress has been delivered".

[2] Saying that the company can 'afford' the expenses only means that it has enough money to pay those expenses.  It does not mean that it will pay those expenses.  Here is an example sentence:

"Your company can afford this payment.  Why are you refusing to pay?"

Therefore, your email needs a different word to say that it will pay the expense of returning the dress.  Here are some options:

"Our company will cover the expenses."

"Our company will pay the expenses."

"Our company will reimburse you for the expenses."

[3] Because you don't know what expenses are involved in returning the dress, it would be better to write, "Our company will cover any expenses."  However your wording was not wrong, so I have not corrected it.

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You've done a good job with this email.  It is very close to natural English.