Learn Hardcore Korean: darimiga gojang nasseoyo.

-가 is the subject marker in Korean. In 다리미가 고장 났어요, the iron (다리미) is the thing that “broke” on its own, so it’s the subject of the intransitive verb 고장 나다 (“to break down”). If you wanted to say “I broke the iron,” you’d use the object marker -를 with 고장 내다 (the transitive counterpart).

What does 고장 났어요 literally mean and how is it formed?

고장 났어요 is the polite-past form of the compound verb 고장나다:

고장 (“breakdown,” a noun)
나다 (“to occur” or “to happen”)
-았/었어요 marks polite past tense.
So literally it’s “a breakdown occurred,” idiomatically “(it) broke.”

Why are 고장 and 났어요 written separately instead of as one word?

Korean often separates a Sino-Korean noun (고장) and the verb (나다) even when they form a single lexical meaning. Writing 고장 났어요 makes the noun–verb boundary clear. You might occasionally see 고장났어요 in very casual texts, but the standard is to split them.

What level of politeness is -었어요, and when should I use it?