Learn Finnish: Polvi on kipeä.

Because the subject, polvi (knee), is a singular, countable noun. When describing a specific, whole, countable thing with the verb olla (to be), the adjective stays in the nominative case. If the subject were an uncountable mass noun (like water or coffee), you would use the partitive.

How do we know it means "the knee" instead of "a knee" if Finnish doesn't have articles?

In Finnish, word order usually handles this. When a noun like polvi is the subject at the very beginning of a basic sentence, it is treated as definite (the knee).

Could I use the essive case here and say Polvi on kipeänä?

You can, but it emphasizes that the soreness is a temporary state. Polvi on kipeä is the standard way to simply say the knee is sore. Using the essive (kipeänä) sounds a bit more like "the knee is currently acting up" or "is in a state of being sore."

Can kipeä be used for both physical pain and being sick?

Yes, it is very versatile. Here it means "sore" or "painful" because it refers to a body part. When used to describe a person, like Hän on kipeä, it translates to "He/She is sick."