Learn Arabic (Modern Standard): كانت فاطمة ممرضة في لندن.

The name فاطمة is a 'diptote' (a special group of words called الممنوع من الصرف). Feminine proper names do not take Tanween. Because Fatima is the subject of the verb كانت, it is in the nominative case. A regular word would take a Tanween Damm (un), but as a diptote, it only takes a single Damma (فاطمةُ Fāṭimatu).

Why does لندن (London) end with a Fatha here even though it comes directly after the preposition في (in)?

Normally, a word following a preposition takes a Kasra (the genitive case). However, لندن is a foreign place name, which makes it a diptote. The defining rule for diptotes is that in the genitive case, they take a Fatha instead of a Kasra. So we say في لندنَ (fī Lundana), never fī Lundani.

If فاطمة and لندن are both diptotes, why does ممرضة (nurse) still get a Tanween?

The word ممرضة is a regular noun, not a proper name or a foreign place name. Most regular nouns take standard case endings, including Tanween. Because it is the predicate of كانت (it tells us what Fatima was), it must be in the accusative case, so it takes the standard Tanween Fath (ممرضةً mumarriḍatan).

Why do we use كانت instead of كان?

The verb كان (to be) must agree in gender with its subject. Since the subject فاطمة is feminine, we add the feminine marker ت (Taa) to the end of the verb, making it كانت (kānat).