* 고장난 것 때문에 (gojangnan geot ttaemune): This translates literally to "because of the broken thing." This is a good general option if you're talking about the reason something happened because something else was broken.
* 고장난 것에 대한 (gojangnan geot-e daehan): This translates to "regarding/about the broken thing." This is suitable if you're talking about something related to the broken item.
* 고장난 것을 위해 (gojangnan geot-eul wihae): This translates to "for the broken thing." This is suitable if you're doing something *to help* repair or replace the broken item.
* 수리 때문에 (suri ttaemune): This translates to "because of the repair." This is suitable if the "broken" aspect is in relation to needing a repair.
* 파손된 것 때문에 (pasondeon geot ttaemune): This translates to "because of the damaged thing." This is a more formal and potentially stronger term than 고장났어요 (gojangnat-seoyo - it's broken).
To choose the best option, please provide the context of your sentence. For example, "For the broken vase" would best be translated as 깨진 화병 때문에 (kkae-jin hwa-byeong ttaemune) (because of the broken vase) or 깨진 화병을 위해 (kkae-jin hwa-byeong-eul wihae) (for the broken vase, if you're doing something for it).