היא סבלנית עם הילדים גם אחרי יום ארוך בעבודה.

Because it describes היא (she), and Hebrew adjectives must agree with the noun or pronoun they describe in gender and number.

• Masculine singular: סבלני
• Feminine singular: סבלנית

So:

• הוא סבלני = he is patient
• היא סבלנית = she is patient

In Hebrew, the verb to be is usually not stated in the present tense.

So:

• היא סבלנית literally looks like she patient
• but it means she is patient

This is completely normal in Hebrew. In past and future, Hebrew does use forms of to be, but in the present it is usually omitted.

עם means with, so עם הילדים means with the children.

In Hebrew, the adjective סבלני / סבלנית commonly goes with עם when you mean being patient toward someone:

• היא סבלנית עם הילדים = she is patient with the children

This matches English quite closely here.

הילדים means the children, while ילדים means children in a general or indefinite sense.

So:

• עם הילדים = with the children
• עם ילדים = with children

The ה at the beginning is the Hebrew definite article, like the in English.

גם usually means also, too, or sometimes even, depending on context.

In this sentence, it gives the sense of:

• also after a long day at work
• or more naturally in English, even after a long day at work

So it emphasizes that her patience continues under difficult circumstances.

אחרי is the preposition meaning after.

• אחרי יום ארוך = after a long day

By contrast, אחר is usually an adjective meaning other or another, depending on context.

So here you need אחרי because the sentence is talking about time: after a long day.

Because יום (day) is a masculine singular noun, and the adjective must agree with it.

• יום = masculine singular
• therefore: ארוך = masculine singular form of long

So:

• יום ארוך = a long day
• שנה ארוכה = a long year

The adjective comes after the noun in Hebrew.

בעבודה means at work or in the workplace.

It is made from:

• ב = in / at
• העבודה = the work / the job / the workplace

When ב combines with ה, they contract:

• ב + העבודה = בעבודה

So יום ארוך בעבודה means a long day at work.

Hebrew has no separate word for a / an.

So:

• יום ארוך can mean a long day
• in another context it could also simply mean long day in a more literal gloss

Whether English needs a or the depends on context and translation.

The word order here is very natural, but Hebrew is somewhat flexible.

Standard order here:

• היא סבלנית עם הילדים גם אחרי יום ארוך בעבודה

You could move parts for emphasis, for example:

• גם אחרי יום ארוך בעבודה, היא סבלנית עם הילדים.

That puts more focus on even after a long day at work.

So the original order is neutral and natural, but other orders are possible.

A common pronunciation would be:

hi savlanit im ha-yeladim gam acharei yom arokh ba-avoda

A rough breakdown:

• היא = hi
• סבלנית = sav-la-NIT
• עם = im
• הילדים = ha-ye-la-DIM
• גם = gam
• אחרי = a-cha-REI
• יום = yom
• ארוך = a-ROKH
• בעבודה = ba-a-vo-DA

In this kind of sentence, סבלנית means patient in the emotional or behavioral sense: calm, tolerant, not easily irritated.

For example:

• היא סבלנית עם הילדים = she is patient with the children

It comes from the same general idea as סבלנות = patience.

So if you want to say She has patience, you could also say:

• יש לה סבלנות

But in your sentence, סבלנית is the adjective patient.

Normally, no. The natural expression is סבלני / סבלנית עם...

So:

• היא סבלנית עם הילדים = correct and natural

Using ל here would sound unnatural for this meaning. If you are describing being patient toward someone, עם is the usual choice.