Mitt knä är svullet, så jag sitter hemma idag.

Because knä is an ett noun in Swedish: ett knä.

Swedish possessives agree with the gender and number of the noun:

• min for singular en-words
• mitt for singular ett-words
• mina for plural nouns

So:

• min arm = my arm
• mitt knä = my knee
• mina ben = my legs

Knä is an ett noun.

Its common forms are:

• indefinite singular: ett knä
• definite singular: knät
• indefinite plural: knän
• definite plural: knäna

In your sentence, mitt knä is singular and indefinite because it means my knee, not the knee.

Because the adjective has to agree with knä, which is an ett noun.

When an adjective comes after är, it still agrees with the noun:

• en-word singular: svullen
• ett-word singular: svullet
• plural: svullna

So:

• Min arm är svullen
• Mitt knä är svullet
• Mina händer är svullna

It is commonly used as an adjective, but it is historically related to the verb svälla, meaning to swell.

For a learner, the most useful thing is to treat svullen / svullet / svullna as an adjective pattern you can use directly:

• ett svullet knä
• knät är svullet

Yes. Swedish often talks about body parts with the definite form in set expressions.

A very common pattern is:

• Jag har ont i knät = I have pain in my knee / My knee hurts

But Mitt knä är svullet is also perfectly natural. It is a bit more direct and specifically describes the knee as swollen, not just painful.

So both are natural, but they focus on slightly different things:

• Jag har ont i knät = the knee hurts
• Mitt knä är svullet = the knee is swollen

Here så means so, therefore, or as a result.

It connects the two ideas:

• Mitt knä är svullet
• så jag sitter hemma idag

So the logic is: My knee is swollen, so I’m staying at home today.

Because så is a coordinating conjunction, and the next clause is still a main clause.

That means the clause can keep normal main-clause word order:

• jag sitter hemma idag

The subject jag comes before the verb sitter.

If you move another element to the front, the word order changes as usual in Swedish:

• Så idag sitter jag hemma
• Idag sitter jag hemma

But in your sentence, så jag sitter hemma idag is the neutral pattern.

Not necessarily.

In Swedish, posture verbs like sitta, stå, and ligga are often used more broadly than in English.
So jag sitter hemma idag can sound like I’m at home today or I’m staying home today, not only I am literally sitting down at home.

In this sentence, it suggests staying at home because of the swollen knee.

A few nearby alternatives are:

• Jag är hemma idag = I’m at home today
• Jag stannar hemma idag = I’m staying home today

Because hemma means at home: a location.

Hem usually expresses direction, meaning home in the sense of to home / homeward.

Compare:

• Jag sitter hemma = I am at home
• Jag går hem = I am going home

So in your sentence, the person is already at home, which is why hemma is used.

Because time expressions are often flexible in Swedish.

The end position is very natural and neutral:

• Jag sitter hemma idag

But you can also move idag earlier for emphasis:

• Idag sitter jag hemma
• Så idag sitter jag hemma

The meaning stays basically the same, but the focus changes a little.

It is acceptable and helpful, but in Swedish it is often not strictly necessary in short coordinated sentences.

So both of these can appear:

• Mitt knä är svullet, så jag sitter hemma idag.
• Mitt knä är svullet så jag sitter hemma idag.

With the comma, the pause and the cause-result relationship are a bit clearer.

No, in Swedish the k is pronounced in knä.

So unlike English knee, Swedish knä starts with an actual k sound.

A rough English-style approximation is kneh, with a clear k at the start.

That is one of the classic differences between English and Swedish spelling/pronunciation for this word.