kakariin no onnanohito ga mousikomisyo no kakikata wo totemo teinei ni setumeisite kureta.

No. Here の is linking two nouns so that 女の人 is being specified as a kind of 係員.

• 係員 = staff member, attendant, clerk
• 女の人 = woman

So 係員の女の人 means something like:

• the woman who was a staff member
• the female staff member
• the woman on staff

In natural English, the female staff member is usually the best fit.

Both can mean woman/female, but they feel a little different.

• 女の人 is a very common everyday way to say woman
• 女性 is more formal or neutral, and often used in writing, announcements, forms, or official descriptions

So 係員の女の人 sounds like normal spoken Japanese.
If you changed it to 女性の係員, it would sound a bit more formal and compact, but the meaning would be similar.

が marks the subject of the sentence: the person who did the explaining.

So:

• 係員の女の人が = the female staff member is the one who...
• 説明してくれた = ...explained (for me/us)

In other words, が tells you who performed the action.

This is another noun + の + noun structure.

• 申込書 = application form
• 書き方 = way of writing / how to write

So 申込書の書き方 means:

• how to fill out the application form
• the way to write the application form
• how to write on the application form

In natural English, how to fill out the application form is often the smoothest translation.

書き方 is made from:

• the verb stem 書き from 書く = to write
• 方 = way / method

So:

• 書く → 書き方 = how to write, way of writing
• 使う → 使い方 = how to use
• 食べる → 食べ方 = how to eat

This pattern is very common in Japanese:

• verb stem + 方 = how to do X / the way of doing X

Because 申込書の書き方 is the thing being explained.

• 説明する = to explain
• what is explained gets marked by を

So:

• 申込書の書き方を説明してくれた
• (someone) explained how to fill out the application form (for me/us)

The particle を marks the direct object of 説明してくれた.

Because 丁寧 is a na-adjective, and when you use a na-adjective adverbially to modify a verb, you usually change it to 〜に.

• 丁寧だ = is polite / careful
• 丁寧に説明する = explain carefully / politely

So:

• とても丁寧に説明してくれた = explained it very carefully/politely

Compare:

• 静かだ = is quiet
• 静かに話す = speak quietly

とても means very, and here it modifies 丁寧に.

So the phrase:

• とても丁寧に

means:

• very carefully
• very politely
• in a very thorough/careful way

It strengthens the manner of the explanation.

This is a very important nuance.

• 説明した = explained
• 説明してくれた = explained for me/us; did me/us the favor of explaining

The verb くれる adds the idea that the action was done for the speaker’s benefit (or for someone in the speaker’s in-group).

So 説明してくれた does not just report the action; it also shows that the speaker felt they received that action.

This often sounds more natural when someone helped you.

The recipient is not stated, but it is understood to be the speaker, or the speaker’s group.

So the sentence implies something like:

• The female staff member kindly explained how to fill out the application form to me/us.

Japanese often leaves this unstated when it is obvious from context.

If you wanted to state it explicitly, you might say:

• 私に説明してくれた = explained it to me

But in many cases, Japanese simply omits 私に.

Because the sentence is talking about a completed past event.

• 説明してくれる = will explain / explains for me
• 説明してくれた = explained for me

So the whole sentence is describing something that already happened:

• a female staff member explained the application form instructions very carefully

Because the sentence is simply introducing or identifying who did the action, so が works naturally.

Using は would shift the feeling a little more toward setting a topic, as in:

• 係員の女の人は… = As for the female staff member...

But the original sentence is more like:

• The female staff member is the one who explained it

So が is a natural choice when focusing on the subject of the action.

It is grammatically plain past, not polite ます style.

• plain: 説明してくれた
• polite: 説明してくれました

But the sentence can still sound perfectly normal in casual narration.
Also, the word 丁寧に describes how the staff member explained, not the politeness level of the speaker’s sentence.

So:

• 丁寧に = carefully / politely
• くれた = plain-style past narration

These are different things.

Yes. In context, 申込書の書き方 often means more than just the physical act of writing letters. It usually means:

• what to write
• where to write it
• how to complete the form properly

So even though 書き方 literally comes from to write, the natural English translation in this context is often how to fill out the application form.

Because の is doing noun-linking work in two different places:

1. 係員の女の人
   = the woman who is a staff member / the female staff member

2. 申込書の書き方
   = the way of writing the application form / how to fill out the application form

Japanese often stacks noun relationships like this instead of using relative clauses or prepositions the way English does.

A fairly literal breakdown is:

• 係員の女の人が
  = the female staff member
• 申込書の書き方を
  = how to fill out the application form
• とても丁寧に
  = very carefully
• 説明してくれた
  = explained for me/us

So literally:

The female staff member explained to me/us, very carefully, how to fill out the application form.

That is a good way to see how the Japanese pieces fit together, even if the final English translation might be smoother.