By definition, malformed XML isn't XML. It's a text file that looks like XML to us, but not to normal parsers.
You can try to write one yourself. Good luck with that as it's not going to be easy.
Or, you can try using this:
GitHub - XmlParser[
^] parser. It's supposed to be fault-tolerant, but I have no idea how extensive that tolerance is.
I think the bigger problem is going to be how you identify which parts of the XML are supposed to be exported to the JSON file. It's not entirely clear how you identify what is valid data and what isn't from the example you posted.