なかなかお手紙を出せずにごめんなさいって英語でなんて言うの？

「手紙を出す」と短く言うには「書く」と言う意味の "write" の過去分詞、"written" を使います。
「なかなか」と英語では直訳がないのですが、"lately" や "recently"　を使い、「最近手紙を送ってなくてごめんなさい」"Sorry I did not write to you recently" と言う風に表現できます。

These are just two examples of apologising for not writing for a while. The first is a more formal apology, whilst the second is more casual, and is used to a good friend.
These two examples are not usually sentences within themselves, but are beginnings of sentences which are normally followed by a reason. For example:-

My apologies for not writing to you sooner, but I have been recovering from a knee operation.
or,
I am sorry that I haven't written to you lately as I have just returned from an extended holiday overseas.

When you are writing a letter to your friend abroad and want to express that you are sorry for not having written them for a while; then you may express this in the following ways:
-I am sorry I have not written to you lately.
-I would like to apologize for not writing sooner.

The best way to start a letter to someone is by apologising. Especially if we want to convey that we it is our fault that we haven't contacted the other person for awhile, and we were supposed to do so.
In the second sentence "as of late" is another way of saying "lately"

You can choose to highlight how long it has been and ask for forgiveness for not writing sooner. Or offer a sincere apology for taking so long to respond. You can use any of the above examples and it will be sincere and effective.

When something takes a while and you want to apologize, you can say, "sorry it's been so long since..." For example, "Sorry it's been so long since I last called you." You can also say "sorry it took so long to..." For example, "Sorry it took so long to have your order ready." You can also say, "I apologize" instead of "Sorry I..." as a more formal and sincere manner of communicating that you feel bad. Thanks!

To express that you feel bad about not writing to your friend, you can use the sentences above.
You can also explain your circumstances to your friend.
Perhaps you have been out of town or you have been really busy at work or school and therefore writing was not easy.

"I miss you so much, I am so sorry for my silence, I have not been able to write recently because I have been writing exams."
"I have been so sick and I didn't get the chance to write to you, please forgive me."

As you are writing to a friend, we tend to use 'sorry' by way of expressing an apology because it is more informal.
We might say:
- I'm sorry I've not written for a while, ...
- I'm sorry that I've not been in touch for a while, ...
...before then continuing on to provide a reason or explanation for the delay e.g. I've been very busy with work/studies etc.
You might also use:
- I hope that you are well, apologies for not writing sooner/before now but...

Situations where this word/phrase is used: social, informal, communications with friends
Type of word/phrase: standard language
Example of this word/phrase:
"So sorry I haven't been in touch recently - but I've just a baby and moved home!"

1. I do apologize for not having kept in touch.
   To "keep in touch" with someone means to remain in
   contact via snail mail (posted letters), email, cell phone
   or social media.
   Example: Please stay/keep in touch while you're abroad.

2. I'm sorry for my prolonged silence.
   "A prolonged silence" means that you have not been writing
   or chatting to your friend for a while and the word "prolonged"
   here specificaly means longer than usual. This refers to the
   fact that you have not been out of touch for this length of
   time before.

Example: After a prolonged length of time, the pupil eventually
responded to the teacher's question.

Here is an example of this sentence being used in context: I have been so busy lately! Sorry I haven't written to you in such a long time, I have just been really caught up with work and looking after my family. How have you been any way? We have so much to catch up on!

If you would like to express your apologies to someone for writing back so late to them, you can say something like "I am sorry for getting back to you so late..." or "I'm sorry it took me so long to reply to you but...". These are some polite and nice ways to say this to someone.