למרות שהמזגן עדיין מקולקל, אני שמחה שהטכנאית כבר יודעת מה הבעיה.

למרות ש־ means although / even though.

The word למרות by itself is more like despite / in spite of and is often followed by a noun:

• למרות הגשם = despite the rain

But when Hebrew wants to say although before a full clause, it commonly uses למרות ש־:

• למרות שהמזגן מקולקל = although the air conditioner is broken

The ש־ is a linking word, roughly like that here.

ה־ is the Hebrew definite article, meaning the.

So:

• מזגן = an air conditioner / air conditioner
• המזגן = the air conditioner

In this sentence, the speaker means a specific air conditioner, so המזגן is the natural form.

מקולקל means broken, out of order, or not working properly.

Here it behaves like an adjective describing המזגן. Since מזגן is masculine singular, the adjective is also masculine singular:

• masculine singular: מקולקל
• feminine singular: מקולקלת
• masculine plural: מקולקלים
• feminine plural: מקולקלות

So המזגן מקולקל = the air conditioner is broken.

In Hebrew, the verb to be is usually omitted in the present tense.

So Hebrew says:

• המזגן מקולקל
  literally: the air conditioner broken

But it means:

• the air conditioner is broken

This is very normal Hebrew. The same thing happens later in מה הבעיה:

• literally: what the problem
• meaning: what the problem is / what the problem is

In past or future, Hebrew does use forms of to be.

עדיין means still.

So:

• המזגן עדיין מקולקל = the air conditioner is still broken

It usually appears before the adjective or phrase it affects, though Hebrew word order can be somewhat flexible. In this sentence, its position is very natural and standard.

No. שמח / שמחה agrees with the speaker's gender.

So:

• a woman says אני שמחה = I am glad
• a man says אני שמח = I am glad

Hebrew often marks gender in adjectives and present-tense verb forms, even in the first person.

Here ש־ means that.

So:

• אני שמחה שהטכנאית כבר יודעת...
  = I’m glad that the technician already knows...

Hebrew often attaches ש־ directly to the next word, instead of writing it separately. So שהטכנאית is really:

• ש + הטכנאית

This kind of ש־ is very common after verbs and adjectives of emotion, thought, speech, and knowledge.

Because הטכנאית is feminine: the female technician.

Compare:

• טכנאי = male technician
• טכנאית = female technician

Since the sentence is talking about a woman, Hebrew uses the feminine noun. The ה־ at the beginning still means the, so:

• הטכנאית = the female technician

Because it must agree with הטכנאית, which is feminine singular.

The verb is from לדעת = to know. In the present tense:

• יודע = he knows
• יודעת = she knows
• יודעים = they know (masculine/mixed)
• יודעות = they know (feminine)

So:

• הטכנאית כבר יודעת = the female technician already knows

כבר means already.

So:

• הטכנאית כבר יודעת = the technician already knows

Using עדיין and כבר together creates a nice contrast:

• the air conditioner is still broken
• but the technician already knows what the problem is

That contrast is a very natural thing to express in Hebrew.

מה הבעיה is the normal everyday Hebrew way to say what the problem is or what the problem is.

Because Hebrew usually omits is in the present tense, מה הבעיה literally looks like:

• what the problem

But it means:

• what the problem is

After יודעת, it works as an embedded question:

• יודעת מה הבעיה = knows what the problem is

A more formal form like מהי הבעיה also exists, but מה הבעיה is much more common in everyday speech.

Yes, but then it usually needs a noun phrase, not a full clause.

For example:

• למרות החום = despite the heat
• למרות התקלה = despite the malfunction

But in your sentence, what follows is a whole clause:

• המזגן עדיין מקולקל

So Hebrew naturally uses:

• למרות שהמזגן עדיין מקולקל = although the air conditioner is still broken

That is why ש־ is needed here.