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The yellow part where you charge the Sony SRS-X55 is broken. How can I replace this?

Darla Dang that is your DC in jack. It is soldered to the I/O board. Remove that board by using this guide Then check if you can unsolder it and find a replacement based on the dimension from places like mouser. com or digikey.com etc. Not a terrible complicated repair but it does require some soldering skills. You can get the jack connectors from here https://ru.aliexpress.com/item/202656625