Die Wunde an meinem Knie sieht klein aus, aber die Ärztin will sie trotzdem untersuchen.

Because Wunde is a feminine noun in German: die Wunde.

In this sentence, Die Wunde is also the subject, so it is in the nominative case.

• die Wunde = the wound

A learner usually just has to memorize the gender with the noun:

• die Wunde
• die Ärztin
• das Knie

Here, an means something like on / at the knee.

German often uses an with body parts for things attached to or located on them:

• eine Wunde an meinem Knie = a wound on my knee
• Schmerzen am Rücken = pain in/on the back

It is meinem Knie because an takes the dative here when it describes a location rather than movement.

So:

• an meinem Knie = on my knee

Because the preposition an is using the dative case here.

The noun Knie is neuter (das Knie), and the possessive mein- has to take the correct dative ending:

• nominative: mein Knie
• dative: meinem Knie

So:

• an meinem Knie = on my knee

Because the verb here is aussehen, which means to look / to appear.

aussehen is a separable verb:

• infinitive: aussehen
• in a main clause: sieht ... aus

So:

• Die Wunde sieht klein aus. = The wound looks small.

The verb is split:

• sieht goes in the normal verb position
• aus goes to the end of the clause

Because klein is being used as a predicate adjective, not directly before a noun.

Compare:

• eine kleine Wunde = a small wound
  → adjective before a noun, so it gets an ending

• Die Wunde ist klein. = The wound is small
  → predicate adjective, so no ending

• Die Wunde sieht klein aus. = The wound looks small
  → also predicate adjective, so no ending

This is normal German main-clause word order.

The finite verb must be in the second position:

• die Ärztin will ...

Then the rest of the sentence follows, and the infinitive goes to the end because of the modal verb will:

• will sie trotzdem untersuchen

Also, aber does not force inversion. It works like English but.

So:

• ..., aber die Ärztin will ... not
• ..., aber will die Ärztin ...

Because will is a modal verb.

With modal verbs in German, the conjugated modal goes in the usual position, and the main verb stays in the infinitive at the end:

• Die Ärztin will sie untersuchen.
• Ich kann Deutsch sprechen.
• Wir müssen jetzt gehen.

So here:

• will = finite/modal verb
• untersuchen = infinitive at the end

trotzdem means nevertheless / anyway / even so.

It shows contrast with the first clause:

• Die Wunde ... sieht klein aus = The wound looks small
• aber die Ärztin will sie trotzdem untersuchen = but the doctor still wants to examine it

So the idea is: It looks small, but despite that, the doctor wants to examine it.

Here, sie refers to die Wunde.

So:

• die Wunde = feminine singular noun
• sie = it in English, referring to that feminine noun

Even though sie can also mean she or they, the context makes it clear that it means the wound here.

So:

• die Ärztin will sie untersuchen = the doctor wants to examine it

Because German pronouns follow the grammatical gender of the noun, not the natural gender or the English word it.

Since Wunde is feminine:

• die Wunde → sie

Examples:

• der Tisch → er
• die Wunde → sie
• das Knie → es

So even though English says it, German uses:

• sie, because Wunde is feminine

Ärztin means female doctor.

It is formed from Arzt (male doctor / doctor) plus the feminine ending -in, and in this word the vowel changes too:

• der Arzt = male doctor / doctor
• die Ärztin = female doctor

So the sentence specifically says the doctor is a woman.

It is in the accusative case because it is the direct object of untersuchen.

Ask:

• What does the doctor want to examine?
• sie = the wound

So:

• die Ärztin = subject
• will untersuchen = verb phrase
• sie = direct object, therefore accusative

For feminine singular, the accusative form is still sie, so it looks the same as the nominative pronoun.

Yes, you may hear that, and it can sound natural too. But an meinem Knie is very common for something located on a body part.

A rough distinction is:

• an meinem Knie = on/at my knee, attached to that area
• auf meinem Knie = physically on top of my knee

With wounds, scars, pain, and similar things, German often prefers an. So Die Wunde an meinem Knie is a very natural choice.